Theorem ipndxnmulrndx

Description: The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca . (Contributed by AV, 29-Oct-2024)

	Ref	Expression
Assertion	ipndxnmulrndx	⊢ ( ·𝑖 ‘ ndx ) ≠ ( .r ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		3re	⊢ 3 ∈ ℝ
2		3lt8	⊢ 3 < 8
3	1 2	gtneii	⊢ 8 ≠ 3
4		ipndx	⊢ ( ·𝑖 ‘ ndx ) = 8
5		mulrndx	⊢ ( .r ‘ ndx ) = 3
6	4 5	neeq12i	⊢ ( ( ·𝑖 ‘ ndx ) ≠ ( .r ‘ ndx ) ↔ 8 ≠ 3 )
7	3 6	mpbir	⊢ ( ·𝑖 ‘ ndx ) ≠ ( .r ‘ ndx )