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Mirrors > Home > MPE Home > Th. List > Mathboxes > mod0mul | Structured version Visualization version GIF version |
Description: If an integer is 0 modulo a positive integer, this integer must be the product of another integer and the modulus. (Contributed by AV, 7-Jun-2020.) |
Ref | Expression |
---|---|
mod0mul | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 12066 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | nnrp 12483 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
3 | mod0 13335 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → ((𝐴 mod 𝑁) = 0 ↔ (𝐴 / 𝑁) ∈ ℤ)) | |
4 | 1, 2, 3 | syl2an 599 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 ↔ (𝐴 / 𝑁) ∈ ℤ)) |
5 | simpr 488 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) → (𝐴 / 𝑁) ∈ ℤ) | |
6 | oveq1 7177 | . . . . . 6 ⊢ (𝑥 = (𝐴 / 𝑁) → (𝑥 · 𝑁) = ((𝐴 / 𝑁) · 𝑁)) | |
7 | 6 | eqeq2d 2749 | . . . . 5 ⊢ (𝑥 = (𝐴 / 𝑁) → (𝐴 = (𝑥 · 𝑁) ↔ 𝐴 = ((𝐴 / 𝑁) · 𝑁))) |
8 | 7 | adantl 485 | . . . 4 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) ∧ 𝑥 = (𝐴 / 𝑁)) → (𝐴 = (𝑥 · 𝑁) ↔ 𝐴 = ((𝐴 / 𝑁) · 𝑁))) |
9 | zcn 12067 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
10 | 9 | adantr 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ) |
11 | nncn 11724 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
12 | 11 | adantl 485 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ) |
13 | nnne0 11750 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
14 | 13 | adantl 485 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0) |
15 | 10, 12, 14 | divcan1d 11495 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / 𝑁) · 𝑁) = 𝐴) |
16 | 15 | adantr 484 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) → ((𝐴 / 𝑁) · 𝑁) = 𝐴) |
17 | 16 | eqcomd 2744 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) → 𝐴 = ((𝐴 / 𝑁) · 𝑁)) |
18 | 5, 8, 17 | rspcedvd 3529 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁)) |
19 | 18 | ex 416 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / 𝑁) ∈ ℤ → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁))) |
20 | 4, 19 | sylbid 243 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∃wrex 3054 (class class class)co 7170 ℂcc 10613 ℝcr 10614 0cc0 10615 · cmul 10620 / cdiv 11375 ℕcn 11716 ℤcz 12062 ℝ+crp 12472 mod cmo 13328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fl 13253 df-mod 13329 |
This theorem is referenced by: m1modmmod 45401 |
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