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Mirrors > Home > MPE Home > Th. List > smndex1iidm | Structured version Visualization version GIF version |
Description: The modulo function 𝐼 is idempotent. (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 |
---|---|
smndex1iidm | ⊢ (𝐼 ∘ 𝐼) = 𝐼 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 11894 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ) | |
2 | smndex1ibas.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
3 | nnrp 12388 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 𝑁 ∈ ℝ+ |
5 | modabs2 13268 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → ((𝑦 mod 𝑁) mod 𝑁) = (𝑦 mod 𝑁)) | |
6 | 1, 4, 5 | sylancl 589 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → ((𝑦 mod 𝑁) mod 𝑁) = (𝑦 mod 𝑁)) |
7 | 6 | eqcomd 2804 | . . 3 ⊢ (𝑦 ∈ ℕ0 → (𝑦 mod 𝑁) = ((𝑦 mod 𝑁) mod 𝑁)) |
8 | 7 | mpteq2ia 5121 | . 2 ⊢ (𝑦 ∈ ℕ0 ↦ (𝑦 mod 𝑁)) = (𝑦 ∈ ℕ0 ↦ ((𝑦 mod 𝑁) mod 𝑁)) |
9 | smndex1ibas.i | . . 3 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
10 | oveq1 7142 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑁) = (𝑦 mod 𝑁)) | |
11 | 10 | cbvmptv 5133 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) = (𝑦 ∈ ℕ0 ↦ (𝑦 mod 𝑁)) |
12 | 9, 11 | eqtri 2821 | . 2 ⊢ 𝐼 = (𝑦 ∈ ℕ0 ↦ (𝑦 mod 𝑁)) |
13 | nn0z 11993 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
14 | 13 | anim2i 619 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℕ0) → (𝑁 ∈ ℕ ∧ 𝑦 ∈ ℤ)) |
15 | 14 | ancomd 465 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℕ0) → (𝑦 ∈ ℤ ∧ 𝑁 ∈ ℕ)) |
16 | zmodcl 13254 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑦 mod 𝑁) ∈ ℕ0) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℕ0) → (𝑦 mod 𝑁) ∈ ℕ0) |
18 | 12 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐼 = (𝑦 ∈ ℕ0 ↦ (𝑦 mod 𝑁))) |
19 | 9 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))) |
20 | oveq1 7142 | . . . 4 ⊢ (𝑥 = (𝑦 mod 𝑁) → (𝑥 mod 𝑁) = ((𝑦 mod 𝑁) mod 𝑁)) | |
21 | 17, 18, 19, 20 | fmptco 6868 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐼 ∘ 𝐼) = (𝑦 ∈ ℕ0 ↦ ((𝑦 mod 𝑁) mod 𝑁))) |
22 | 2, 21 | ax-mp 5 | . 2 ⊢ (𝐼 ∘ 𝐼) = (𝑦 ∈ ℕ0 ↦ ((𝑦 mod 𝑁) mod 𝑁)) |
23 | 8, 12, 22 | 3eqtr4ri 2832 | 1 ⊢ (𝐼 ∘ 𝐼) = 𝐼 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 ℝ+crp 12377 mod cmo 13232 EndoFMndcefmnd 18025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 |
This theorem is referenced by: smndex1mgm 18064 smndex1mndlem 18066 |
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