Theorem dsndxnplusgndx

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

	Ref	Expression
Assertion	dsndxnplusgndx	⊢ ( dist ‘ ndx ) ≠ ( +g ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		2re	⊢ 2 ∈ ℝ
2		1nn	⊢ 1 ∈ ℕ
3		2nn0	⊢ 2 ∈ ℕ0
4		2lt10	⊢ 2 < ; 1 0
5	2 3 3 4	declti	⊢ 2 < ; 1 2
6	1 5	gtneii	⊢ ; 1 2 ≠ 2
7		dsndx	⊢ ( dist ‘ ndx ) = ; 1 2
8		plusgndx	⊢ ( +g ‘ ndx ) = 2
9	7 8	neeq12i	⊢ ( ( dist ‘ ndx ) ≠ ( +g ‘ ndx ) ↔ ; 1 2 ≠ 2 )
10	6 9	mpbir	⊢ ( dist ‘ ndx ) ≠ ( +g ‘ ndx )