IBM Article: miners to use resource intense calcs to verify tx?

[pebwindkraft]

In the IBM article on

https://www.ibm.com/blogs/blockchain/2017/05/the-difference-between-bitcoin-and-blockchain-for-business/?social_post=1099763016&linkId=43090982

it says:

How does the Bitcoin blockchain work?

The Bitcoin blockchain in its simplest form is a database or ledger comprised of Bitcoin transaction records. However, because this database is distributed across a peer-to-peer network and is without a central authority, network participants must agree on the validity of transactions before they can be recorded. This agreement, which is known as “consensus,” is achieved through a process called “mining.”

After someone uses Bitcoins, miners engage in complex, resource-intense computational equations to verify the legitimacy of the transaction.

I was wondering if this statement is correct. Isn't every full node checking conformity to the rules of a transaction? And for the miners the ressource intensive thing is to then find the nonce?
What would the mempool be in this scenario?

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[pebwindkraft]

I'll answer based on the doc in Andreas' book "mastering bitcoin":

Each full node in the bitcoin network verifies every transaction against a long checklist of criteria. This can be time consuming, when a specially crafted tx is used. But usually tx are checked within milli seconds.

Mining nodes will also validate transactions, they are then added to the mempool, where transactions wait until they can be included into a block. The mining node will aggregate these transactions into a candidate block. The block becomes valid only if the miner succeeds in finding a solution to the proof-of-work algorithm. Usually a hardware mining rig is used to find a solution to the proof-of-work algorithm that makes the block valid. The proof of work is the intensive calculation step.
Once the candidate block is filled with transactions, the hash of this block's header is calculated, until the hash is less than a predefined target. At the current difficulty in the bitcoin network, miners have to try quadrillions of times before finding a nonce that results in a low enough block header hash.
And this is the calculation/computing power intensive step. Not validating transactions.