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| Description: Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) | 
| Ref | Expression | 
|---|---|
| 0dvds | ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0z 12626 | . . . 4 ⊢ 0 ∈ ℤ | |
| 2 | divides 16293 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) | 
| 4 | zcn 12620 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
| 5 | 4 | mul01d 11461 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 0) = 0) | 
| 6 | eqtr2 2760 | . . . . . 6 ⊢ (((𝑛 · 0) = 𝑁 ∧ (𝑛 · 0) = 0) → 𝑁 = 0) | |
| 7 | 5, 6 | sylan2 593 | . . . . 5 ⊢ (((𝑛 · 0) = 𝑁 ∧ 𝑛 ∈ ℤ) → 𝑁 = 0) | 
| 8 | 7 | ancoms 458 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 · 0) = 𝑁) → 𝑁 = 0) | 
| 9 | 8 | rexlimiva 3146 | . . 3 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁 → 𝑁 = 0) | 
| 10 | 3, 9 | biimtrdi 253 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 → 𝑁 = 0)) | 
| 11 | dvds0 16310 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
| 12 | 1, 11 | ax-mp 5 | . . 3 ⊢ 0 ∥ 0 | 
| 13 | breq2 5146 | . . 3 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0)) | |
| 14 | 12, 13 | mpbiri 258 | . 2 ⊢ (𝑁 = 0 → 0 ∥ 𝑁) | 
| 15 | 10, 14 | impbid1 225 | 1 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 class class class wbr 5142 (class class class)co 7432 0cc0 11156 · cmul 11161 ℤcz 12615 ∥ cdvds 16291 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-neg 11496 df-z 12616 df-dvds 16292 | 
| This theorem is referenced by: fsumdvds 16346 dvdsabseq 16351 dfgcd2 16584 dvdssq 16605 rpdvds 16698 pcdvdstr 16915 pc2dvds 16918 mndodcongi 19562 oddvdsnn0 19563 oddvds 19566 odmulgeq 19576 odf1 19581 odf1o1 19591 gexdvds 19603 gexnnod 19607 torsubg 19873 ablsimpgfindlem1 20128 ablsimpgfindlem2 20129 znf1o 21571 dvdsexpnn0 42374 jm2.19 43010 nzss 44341 | 
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