Theorem scandxnbasendx

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

	Ref	Expression
Assertion	scandxnbasendx	⊢ ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		1lt5	⊢ 1 < 5
3	1 2	gtneii	⊢ 5 ≠ 1
4		scandx	⊢ ( Scalar ‘ ndx ) = 5
5		basendx	⊢ ( Base ‘ ndx ) = 1
6	4 5	neeq12i	⊢ ( ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx ) ↔ 5 ≠ 1 )
7	3 6	mpbir	⊢ ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx )