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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 12323 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | nnrp 12741 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
3 | mod0 13596 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → ((𝐴 mod 𝑁) = 0 ↔ (𝐴 / 𝑁) ∈ ℤ)) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 ↔ (𝐴 / 𝑁) ∈ ℤ)) |
5 | simpr 485 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) → (𝐴 / 𝑁) ∈ ℤ) | |
6 | oveq1 7282 | . . . . . 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 482 | . . . 4 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) ∧ 𝑥 = (𝐴 / 𝑁)) → (𝐴 = (𝑥 · 𝑁) ↔ 𝐴 = ((𝐴 / 𝑁) · 𝑁))) |
9 | zcn 12324 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
10 | 9 | adantr 481 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ) |
11 | nncn 11981 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
12 | 11 | adantl 482 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ) |
13 | nnne0 12007 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
14 | 13 | adantl 482 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0) |
15 | 10, 12, 14 | divcan1d 11752 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / 𝑁) · 𝑁) = 𝐴) |
16 | 15 | adantr 481 | . . . . 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 3563 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / 𝑁) ∈ ℤ) → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁)) |
19 | 18 | ex 413 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / 𝑁) ∈ ℤ → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁))) |
20 | 4, 19 | sylbid 239 | 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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 · cmul 10876 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℝ+crp 12730 mod cmo 13589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fl 13512 df-mod 13590 |
This theorem is referenced by: m1modmmod 45867 |
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