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Metamath Proof Explorer

Theorem tsetndxnmulrndx

Description: The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024)

		Ref	Expression
	Assertion	tsetndxnmulrndx	⊢ ( TopSet ‘ ndx ) ≠ ( ._r ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		3re	⊢ 3 ∈ ℝ
2		3lt9	⊢ 3 < 9
3	1 2	gtneii	⊢ 9 ≠ 3
4		tsetndx	⊢ ( TopSet ‘ ndx ) = 9
5		mulrndx	⊢ ( ._r ‘ ndx ) = 3
6	4 5	neeq12i	⊢ ( ( TopSet ‘ ndx ) ≠ ( ._r ‘ ndx ) ↔ 9 ≠ 3 )
7	3 6	mpbir	⊢ ( TopSet ‘ ndx ) ≠ ( ._r ‘ ndx )