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Metamath Proof Explorer

Theorem starvndxnmulrndx

Description: The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv . (Contributed by AV, 18-Oct-2024)

		Ref	Expression
	Assertion	starvndxnmulrndx	⊢ ( *_𝑟 ‘ ndx ) ≠ ( ._r ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		3re	⊢ 3 ∈ ℝ
2		3lt4	⊢ 3 < 4
3	1 2	gtneii	⊢ 4 ≠ 3
4		starvndx	⊢ ( *_𝑟 ‘ ndx ) = 4
5		mulrndx	⊢ ( ._r ‘ ndx ) = 3
6	4 5	neeq12i	⊢ ( ( *_𝑟 ‘ ndx ) ≠ ( ._r ‘ ndx ) ↔ 4 ≠ 3 )
7	3 6	mpbir	⊢ ( *_𝑟 ‘ ndx ) ≠ ( ._r ‘ ndx )