Theorem tsetndxnplusgndx

Description: The slot for the topology is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgtset . (Contributed by AV, 18-Oct-2024)

	Ref	Expression
Assertion	tsetndxnplusgndx	⊢ ( TopSet ‘ ndx ) ≠ ( +g ‘ ndx )

Proof

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