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Theorem smndex1ibas 18851

Description: The modulo function 𝐼 is an endofunction on ℕ₀. (Contributed by AV, 12-Feb-2024.)

**Hypotheses**
Ref	Expression
smndex1ibas.m	⊢ 𝑀 = (EndoFMnd‘ℕ₀)
smndex1ibas.n	⊢ 𝑁 ∈ ℕ
smndex1ibas.i	⊢ 𝐼 = (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁))

**Assertion**
Ref	Expression
smndex1ibas	⊢ 𝐼 ∈ (Base‘𝑀)

**Proof of Theorem smndex1ibas**
Step	Hyp	Ref	Expression
1		eqid 2728	. . . 4 ⊢ (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)) = (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁))
2		nn0z 12613	. . . . 5 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ ℤ)
3		smndex1ibas.n	. . . . . 6 ⊢ 𝑁 ∈ ℕ
4	3	a1i 11	. . . . 5 ⊢ (𝑥 ∈ ℕ₀ → 𝑁 ∈ ℕ)
5	2, 4	zmodcld 13889	. . . 4 ⊢ (𝑥 ∈ ℕ₀ → (𝑥 mod 𝑁) ∈ ℕ₀)
6	1, 5	fmpti 7122	. . 3 ⊢ (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)):ℕ₀⟶ℕ₀
7		nn0ex 12508	. . . 4 ⊢ ℕ₀ ∈ V
8	7, 7	elmap 8889	. . 3 ⊢ ((𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)) ∈ (ℕ₀ ↑_m ℕ₀) ↔ (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)):ℕ₀⟶ℕ₀)
9	6, 8	mpbir 230	. 2 ⊢ (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)) ∈ (ℕ₀ ↑_m ℕ₀)
10		smndex1ibas.i	. 2 ⊢ 𝐼 = (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁))
11		smndex1ibas.m	. . 3 ⊢ 𝑀 = (EndoFMnd‘ℕ₀)
12		eqid 2728	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
13	11, 12	efmndbas 18822	. 2 ⊢ (Base‘𝑀) = (ℕ₀ ↑_m ℕ₀)
14	9, 10, 13	3eltr4i 2842	1 ⊢ 𝐼 ∈ (Base‘𝑀)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 ↦ cmpt 5231 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ↑_m cmap 8844 ℕcn 12242 ℕ₀cn0 12502 mod cmo 13866 Basecbs 17179 EndoFMndcefmnd 18819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fl 13789 df-mod 13867 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-tset 17251 df-efmnd 18820

This theorem is referenced by: smndex1basss 18856 smndex1mgm 18858 smndex1mndlem 18860 smndex1id 18862

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