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Mirrors > Home > MPE Home > Th. List > zmodidfzoimp | Structured version Visualization version GIF version |
Description: Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.) |
Ref | Expression |
---|---|
zmodidfzoimp | ⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo0 13691 | . . 3 ⊢ (𝑀 ∈ (0..^𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑀 < 𝑁)) | |
2 | nn0z 12599 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ) | |
3 | 2 | anim1i 614 | . . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ)) |
4 | 3 | 3adant3 1130 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑀 < 𝑁) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ)) |
5 | 1, 4 | sylbi 216 | . 2 ⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ)) |
6 | zmodidfzo 13883 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀 ↔ 𝑀 ∈ (0..^𝑁))) | |
7 | 6 | biimprd 247 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)) |
8 | 5, 7 | mpcom 38 | 1 ⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 0cc0 11124 < clt 11264 ℕcn 12228 ℕ0cn0 12488 ℤcz 12574 ..^cfzo 13645 mod cmo 13852 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 |
This theorem is referenced by: modfzo0difsn 13926 modlteq 13928 addmodlteq 13929 cshwidxmodr 14772 cshwidx0 14774 cshweqrep 14789 cshw1 14790 cshwshashlem1 17050 smndex1igid 18841 eucrctshift 30027 cshwrnid 32651 cycpmfv1 32799 cycpmfv2 32800 |
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