Theorem smndex1gid

Description: The composition of a constant function ( GK ) with another endofunction on NN0 results in ( GK ) itself. (Contributed by AV, 14-Feb-2024) Avoid ax-rep . (Revised by GG, 2-Apr-2026)

	Ref	Expression
Hypotheses	smndex1ibas.m	$$⊢M=\mathrm{EndoFMnd}\left({\mathbb{N}}_0\right)$$
	smndex1ibas.n	$$⊢N\in \mathbb{N}$$
	smndex1ibas.i	$$⊢I=\left(x\in {\mathbb{N}}_0⟼x\mathrm{mod}N\right)$$
	smndex1ibas.g	$$⊢G=\left(n\in \left(0..^N\right)⟼\left(x\in {\mathbb{N}}_0⟼n\right)\right)$$
Assertion	smndex1gid	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to G\left(K\right)\circ F=G\left(K\right)$$

Proof

Step	Hyp	Ref	Expression
1		smndex1ibas.m	$$⊢M=\mathrm{EndoFMnd}\left({\mathbb{N}}_0\right)$$
2		smndex1ibas.n	$$⊢N\in \mathbb{N}$$
3		smndex1ibas.i	$$⊢I=\left(x\in {\mathbb{N}}_0⟼x\mathrm{mod}N\right)$$
4		smndex1ibas.g	$$⊢G=\left(n\in \left(0..^N\right)⟼\left(x\in {\mathbb{N}}_0⟼n\right)\right)$$
5		id	$$⊢n=K\to n=K$$
6	5	mpteq2dv	$$⊢n=K\to \left(x\in {\mathbb{N}}_0⟼n\right)=\left(x\in {\mathbb{N}}_0⟼K\right)$$
7		fconstmpt	$$⊢{\mathbb{N}}_0\times \left\{K\right\}=\left(x\in {\mathbb{N}}_0⟼K\right)$$
8		nn0ex	$$⊢{\mathbb{N}}_0\in \mathrm{V}$$
9		snex	$$⊢\left\{K\right\}\in \mathrm{V}$$
10	8 9	xpex	$$⊢{\mathbb{N}}_0\times \left\{K\right\}\in \mathrm{V}$$
11	7 10	eqeltrri	$$⊢\left(x\in {\mathbb{N}}_0⟼K\right)\in \mathrm{V}$$
12	6 4 11	fvmpt	$$⊢K\in \left(0..^N\right)\to G\left(K\right)=\left(x\in {\mathbb{N}}_0⟼K\right)$$
13	12	adantl	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to G\left(K\right)=\left(x\in {\mathbb{N}}_0⟼K\right)$$
14	13	adantr	$$⊢\left(\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\wedge y\in {\mathbb{N}}_0\right)\to G\left(K\right)=\left(x\in {\mathbb{N}}_0⟼K\right)$$
15		eqidd	$$⊢\left(\left(\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\wedge y\in {\mathbb{N}}_0\right)\wedge x=F\left(y\right)\right)\to K=K$$
16		eqid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_M$$
17	1 16	efmndbasf	$$⊢F\in {\mathrm{Base}}_M\to F:{\mathbb{N}}_0⟶{\mathbb{N}}_0$$
18		ffvelcdm	$$⊢\left(F:{\mathbb{N}}_0⟶{\mathbb{N}}_0\wedge y\in {\mathbb{N}}_0\right)\to F\left(y\right)\in {\mathbb{N}}_0$$
19	18	ex	$$⊢F:{\mathbb{N}}_0⟶{\mathbb{N}}_0\to \left(y\in {\mathbb{N}}_0\to F\left(y\right)\in {\mathbb{N}}_0\right)$$
20	17 19	syl	$$⊢F\in {\mathrm{Base}}_M\to \left(y\in {\mathbb{N}}_0\to F\left(y\right)\in {\mathbb{N}}_0\right)$$
21	20	adantr	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to \left(y\in {\mathbb{N}}_0\to F\left(y\right)\in {\mathbb{N}}_0\right)$$
22	21	imp	$$⊢\left(\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\wedge y\in {\mathbb{N}}_0\right)\to F\left(y\right)\in {\mathbb{N}}_0$$
23		simplr	$$⊢\left(\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\wedge y\in {\mathbb{N}}_0\right)\to K\in \left(0..^N\right)$$
24	14 15 22 23	fvmptd	$$⊢\left(\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\wedge y\in {\mathbb{N}}_0\right)\to G\left(K\right)\left(F\left(y\right)\right)=K$$
25	24	mpteq2dva	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to \left(y\in {\mathbb{N}}_0⟼G\left(K\right)\left(F\left(y\right)\right)\right)=\left(y\in {\mathbb{N}}_0⟼K\right)$$
26	1 2 3 4	smndex1gbas	$$⊢K\in \left(0..^N\right)\to G\left(K\right)\in {\mathrm{Base}}_M$$
27	1 16	efmndbasf	$$⊢G\left(K\right)\in {\mathrm{Base}}_M\to G\left(K\right):{\mathbb{N}}_0⟶{\mathbb{N}}_0$$
28	26 27	syl	$$⊢K\in \left(0..^N\right)\to G\left(K\right):{\mathbb{N}}_0⟶{\mathbb{N}}_0$$
29		fcompt	$$⊢\left(G\left(K\right):{\mathbb{N}}_0⟶{\mathbb{N}}_0\wedge F:{\mathbb{N}}_0⟶{\mathbb{N}}_0\right)\to G\left(K\right)\circ F=\left(y\in {\mathbb{N}}_0⟼G\left(K\right)\left(F\left(y\right)\right)\right)$$
30	28 17 29	syl2anr	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to G\left(K\right)\circ F=\left(y\in {\mathbb{N}}_0⟼G\left(K\right)\left(F\left(y\right)\right)\right)$$
31		eqidd	$$⊢x=y\to K=K$$
32	31	cbvmptv	$$⊢\left(x\in {\mathbb{N}}_0⟼K\right)=\left(y\in {\mathbb{N}}_0⟼K\right)$$
33	6 32	eqtrdi	$$⊢n=K\to \left(x\in {\mathbb{N}}_0⟼n\right)=\left(y\in {\mathbb{N}}_0⟼K\right)$$
34		fconstmpt	$$⊢{\mathbb{N}}_0\times \left\{K\right\}=\left(y\in {\mathbb{N}}_0⟼K\right)$$
35	34 10	eqeltrri	$$⊢\left(y\in {\mathbb{N}}_0⟼K\right)\in \mathrm{V}$$
36	33 4 35	fvmpt	$$⊢K\in \left(0..^N\right)\to G\left(K\right)=\left(y\in {\mathbb{N}}_0⟼K\right)$$
37	36	adantl	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to G\left(K\right)=\left(y\in {\mathbb{N}}_0⟼K\right)$$
38	25 30 37	3eqtr4d	$$⊢\left(F\in {\mathrm{Base}}_M\wedge K\in \left(0..^N\right)\right)\to G\left(K\right)\circ F=G\left(K\right)$$