Theorem plendxnvscandx

Description: The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. Formerly part of proof for opsrvsca . (Contributed by AV, 1-Nov-2024)

	Ref	Expression
Assertion	plendxnvscandx	⊢ ( le ‘ ndx ) ≠ ( ·𝑠 ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		6re	⊢ 6 ∈ ℝ
2		6lt10	⊢ 6 < ; 1 0
3	1 2	gtneii	⊢ ; 1 0 ≠ 6
4		plendx	⊢ ( le ‘ ndx ) = ; 1 0
5		vscandx	⊢ ( ·𝑠 ‘ ndx ) = 6
6	4 5	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ↔ ; 1 0 ≠ 6 )
7	3 6	mpbir	⊢ ( le ‘ ndx ) ≠ ( ·𝑠 ‘ ndx )