Theorem plendxnplusgndx

Description: The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgle . (Contributed by AV, 18-Oct-2024)

	Ref	Expression
Assertion	plendxnplusgndx	⊢ ( le ‘ ndx ) ≠ ( +g ‘ ndx )

Proof

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