Theorem dsndxnmulrndx

Description: The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024)

	Ref	Expression
Assertion	dsndxnmulrndx	⊢ ( dist ‘ ndx ) ≠ ( .r ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		3re	⊢ 3 ∈ ℝ
2		1nn	⊢ 1 ∈ ℕ
3		2nn0	⊢ 2 ∈ ℕ0
4		3nn0	⊢ 3 ∈ ℕ0
5		3lt10	⊢ 3 < ; 1 0
6	2 3 4 5	declti	⊢ 3 < ; 1 2
7	1 6	gtneii	⊢ ; 1 2 ≠ 3
8		dsndx	⊢ ( dist ‘ ndx ) = ; 1 2
9		mulrndx	⊢ ( .r ‘ ndx ) = 3
10	8 9	neeq12i	⊢ ( ( dist ‘ ndx ) ≠ ( .r ‘ ndx ) ↔ ; 1 2 ≠ 3 )
11	7 10	mpbir	⊢ ( dist ‘ ndx ) ≠ ( .r ‘ ndx )