Theorem vscandxnscandx

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

	Ref	Expression
Assertion	vscandxnscandx	⊢ ( ·𝑠 ‘ ndx ) ≠ ( Scalar ‘ ndx )

Proof

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