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Metamath Proof Explorer

Theorem scandxnplusgndx

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

Ref Expression
Assertion scandxnplusgndx ( Scalar ‘ ndx ) ≠ ( +g ‘ ndx )

Proof

Step Hyp Ref Expression
1 2re ⊢ 2 ∈ ℝ
2 2lt5 ⊢ 2 < 5
3 1 2 gtneii ⊢ 5 ≠ 2
4 scandx ⊢ ( Scalar ‘ ndx ) = 5
5 plusgndx ⊢ ( +g ‘ ndx ) = 2
6 4 5 neeq12i ⊢ ( ( Scalar ‘ ndx ) ≠ ( +g ‘ ndx ) ↔ 5 ≠ 2 )
7 3 6 mpbir ⊢ ( Scalar ‘ ndx ) ≠ ( +g ‘ ndx )