Does every miner verify blocks that are already in the Blockchain

Does every miner verify blocks that are already in the Blockchain

Once a miner adds a block to the chain, does every other miner have to check that block to make sure that its transactions don't violate any rules?

For instance, if a miner added a block with an invalid transaction, is every other miner expected to check that block (and perhaps compare each transaction in it with every other transaction in every other block, to prevent, say, double spending)?

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