Theorem ipndxnplusgndx

Description: The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024)

	Ref	Expression
Assertion	ipndxnplusgndx	⊢ ( ·𝑖 ‘ ndx ) ≠ ( +g ‘ ndx )

Proof

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