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Mirrors > Home > ILE Home > Th. List > addmodid | 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 | simp2 939 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 ∈ ℕ) | |
2 | 1 | nncnd 8053 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 ∈ ℂ) |
3 | 2 | mulid2d 7137 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (1 · 𝑀) = 𝑀) |
4 | 3 | eqcomd 2086 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 = (1 · 𝑀)) |
5 | 4 | oveq1d 5547 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (𝑀 + 𝐴) = ((1 · 𝑀) + 𝐴)) |
6 | 5 | oveq1d 5547 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = (((1 · 𝑀) + 𝐴) mod 𝑀)) |
7 | 1zzd 8378 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 1 ∈ ℤ) | |
8 | nnq 8718 | . . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ) | |
9 | 8 | 3ad2ant2 960 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 ∈ ℚ) |
10 | simp1 938 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℕ0) | |
11 | 10 | nn0zd 8467 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℤ) |
12 | zq 8711 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
13 | 11, 12 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℚ) |
14 | nn0re 8297 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
15 | 14 | 3ad2ant1 959 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℝ) |
16 | 10 | nn0ge0d 8344 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 0 ≤ 𝐴) |
17 | simp3 940 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 < 𝑀) | |
18 | 0re 7119 | . . . . 5 ⊢ 0 ∈ ℝ | |
19 | nnre 8046 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
20 | 19 | rexrd 7168 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ*) |
21 | 20 | 3ad2ant2 960 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝑀 ∈ ℝ*) |
22 | elico2 8960 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ*) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) | |
23 | 18, 21, 22 | sylancr 405 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) |
24 | 15, 16, 17, 23 | mpbir3and 1121 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → 𝐴 ∈ (0[,)𝑀)) |
25 | mulqaddmodid 9366 | . . 3 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℚ) ∧ (𝐴 ∈ ℚ ∧ 𝐴 ∈ (0[,)𝑀))) → (((1 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) | |
26 | 7, 9, 13, 24, 25 | syl22anc 1170 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → (((1 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) |
27 | 6, 26 | eqtrd 2113 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 ℝ*cxr 7152 < clt 7153 ≤ cle 7154 ℕcn 8039 ℕ0cn0 8288 ℤcz 8351 ℚcq 8704 [,)cico 8913 mod cmo 9324 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-q 8705 df-rp 8735 df-ico 8917 df-fl 9274 df-mod 9325 |
This theorem is referenced by: addmodidr 9375 |
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