Theorem vscandxnplusgndx

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

	Ref	Expression
Assertion	vscandxnplusgndx	⊢ ( ·𝑠 ‘ ndx ) ≠ ( +g ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		2re	⊢ 2 ∈ ℝ
2		2lt6	⊢ 2 < 6
3	1 2	gtneii	⊢ 6 ≠ 2
4		vscandx	⊢ ( ·𝑠 ‘ ndx ) = 6
5		plusgndx	⊢ ( +g ‘ ndx ) = 2
6	4 5	neeq12i	⊢ ( ( ·𝑠 ‘ ndx ) ≠ ( +g ‘ ndx ) ↔ 6 ≠ 2 )
7	3 6	mpbir	⊢ ( ·𝑠 ‘ ndx ) ≠ ( +g ‘ ndx )