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Mirrors > Home > MPE Home > Th. List > addmodid | Structured version Visualization version GIF version |
Description: The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.) |
Ref | Expression |
---|---|
addmodid | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 11028 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ) | |
2 | 1 | mulid2d 10058 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → (1 · 𝑀) = 𝑀) |
3 | 2 | 3ad2ant2 1083 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (1 · 𝑀) = 𝑀) |
4 | 3 | eqcomd 2628 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 = (1 · 𝑀)) |
5 | 4 | oveq1d 6665 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (𝑀 + 𝐴) = ((1 · 𝑀) + 𝐴)) |
6 | 5 | oveq1d 6665 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = (((1 · 𝑀) + 𝐴) mod 𝑀)) |
7 | 1zzd 11408 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 1 ∈ ℤ) | |
8 | nnrp 11842 | . . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ+) | |
9 | 8 | 3ad2ant2 1083 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 ∈ ℝ+) |
10 | nn0re 11301 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
11 | 10 | rexrd 10089 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
12 | 11 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℝ*) |
13 | nn0ge0 11318 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴) | |
14 | 13 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 0 ≤ 𝐴) |
15 | simp3 1063 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 < 𝑀) | |
16 | 0xr 10086 | . . . . 5 ⊢ 0 ∈ ℝ* | |
17 | nnre 11027 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
18 | 17 | rexrd 10089 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ*) |
19 | 18 | 3ad2ant2 1083 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 ∈ ℝ*) |
20 | elico1 12218 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) | |
21 | 16, 19, 20 | sylancr 695 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) |
22 | 12, 14, 15, 21 | mpbir3and 1245 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ (0[,)𝑀)) |
23 | muladdmodid 12710 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ∧ 𝐴 ∈ (0[,)𝑀)) → (((1 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) | |
24 | 7, 9, 22, 23 | syl3anc 1326 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (((1 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) |
25 | 6, 24 | eqtrd 2656 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 ℕ0cn0 11292 ℤcz 11377 ℝ+crp 11832 [,)cico 12177 mod cmo 12668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fl 12593 df-mod 12669 |
This theorem is referenced by: addmodidr 12719 cshwidxn 13555 eucrctshift 27103 ex-mod 27306 |
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