Theorem ipndxnbasendx

Description: The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024)

	Ref	Expression
Assertion	ipndxnbasendx	⊢ ( ·𝑖 ‘ ndx ) ≠ ( Base ‘ ndx )

Proof

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