Theorem basendxneufndx

Description: The slot for the base set is not the slot for the Euclidean function in an extensible structure. (Contributed by Thierry Arnoux, 9-Nov-2026)

	Ref	Expression
Assertion	basendxneufndx	⊢ ( Base ‘ ndx ) ≠ ( EuclF ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		basendxnn	⊢ ( Base ‘ ndx ) ∈ ℕ
2	1	nnrei	⊢ ( Base ‘ ndx ) ∈ ℝ
3		2nn	⊢ 2 ∈ ℕ
4		1nn0	⊢ 1 ∈ ℕ0
5		1lt10	⊢ 1 < ; 1 0
6	3 4 4 5	declti	⊢ 1 < ; 2 1
7		basendx	⊢ ( Base ‘ ndx ) = 1
8		eufndx	⊢ ( EuclF ‘ ndx ) = ; 2 1
9	6 7 8	3brtr4i	⊢ ( Base ‘ ndx ) < ( EuclF ‘ ndx )
10	2 9	ltneii	⊢ ( Base ‘ ndx ) ≠ ( EuclF ‘ ndx )