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Mirrors > Home > MPE Home > Th. List > smndex1ibas | Structured version Visualization version GIF version |
Description: The modulo function 𝐼 is an endofunction on ℕ0. (Contributed by AV, 12-Feb-2024.) |
Ref | Expression |
---|---|
smndex1ibas.m | ⊢ 𝑀 = (EndoFMnd‘ℕ0) |
smndex1ibas.n | ⊢ 𝑁 ∈ ℕ |
smndex1ibas.i | ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) |
Ref | Expression |
---|---|
smndex1ibas | ⊢ 𝐼 ∈ (Base‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
2 | nn0z 12483 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
3 | smndex1ibas.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑁 ∈ ℕ) |
5 | 2, 4 | zmodcld 13752 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → (𝑥 mod 𝑁) ∈ ℕ0) |
6 | 1, 5 | fmpti 7057 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)):ℕ0⟶ℕ0 |
7 | nn0ex 12378 | . . . 4 ⊢ ℕ0 ∈ V | |
8 | 7, 7 | elmap 8768 | . . 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 2738 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
13 | 11, 12 | efmndbas 18641 | . 2 ⊢ (Base‘𝑀) = (ℕ0 ↑m ℕ0) |
14 | 9, 10, 13 | 3eltr4i 2852 | 1 ⊢ 𝐼 ∈ (Base‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ↦ cmpt 5187 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 ↑m cmap 8724 ℕcn 12112 ℕ0cn0 12372 mod cmo 13729 Basecbs 17043 EndoFMndcefmnd 18638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-inf 9338 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-uz 12723 df-rp 12871 df-fz 13380 df-fl 13652 df-mod 13730 df-struct 16979 df-slot 17014 df-ndx 17026 df-base 17044 df-plusg 17106 df-tset 17112 df-efmnd 18639 |
This theorem is referenced by: smndex1basss 18675 smndex1mgm 18677 smndex1mndlem 18679 smndex1id 18681 |
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