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Metamath Proof Explorer

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 )