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	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{EuclF}\left(\mathrm{ndx}\right)$$

Proof

Step	Hyp	Ref	Expression
1		basendxnn	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in \mathbb{N}$$
2	1	nnrei	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in ℝ$$
3		2nn	$$⊢2\in \mathbb{N}$$
4		1nn0	$$⊢1\in {\mathbb{N}}_0$$
5		1lt10	$$⊢1<10$$
6	3 4 4 5	declti	$$⊢1<21$$
7		basendx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
8		eufndx	$$⊢\mathrm{EuclF}\left(\mathrm{ndx}\right)=21$$
9	6 7 8	3brtr4i	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<\mathrm{EuclF}\left(\mathrm{ndx}\right)$$
10	2 9	ltneii	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{EuclF}\left(\mathrm{ndx}\right)$$