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 )