Theorem vscandxnbasendx

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

	Ref	Expression
Assertion	vscandxnbasendx	⊢ ( ·𝑠 ‘ ndx ) ≠ ( Base ‘ ndx )

Proof

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