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Mirrors > Home > ILE Home > Th. List > 0dvds | GIF version |
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 9193 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | divides 11715 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) | |
3 | 1, 2 | mpan 421 | . . 3 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) |
4 | zcn 9187 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
5 | 4 | mul01d 8282 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 0) = 0) |
6 | eqtr2 2183 | . . . . . 6 ⊢ (((𝑛 · 0) = 𝑁 ∧ (𝑛 · 0) = 0) → 𝑁 = 0) | |
7 | 5, 6 | sylan2 284 | . . . . 5 ⊢ (((𝑛 · 0) = 𝑁 ∧ 𝑛 ∈ ℤ) → 𝑁 = 0) |
8 | 7 | ancoms 266 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 · 0) = 𝑁) → 𝑁 = 0) |
9 | 8 | rexlimiva 2576 | . . 3 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁 → 𝑁 = 0) |
10 | 3, 9 | syl6bi 162 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 → 𝑁 = 0)) |
11 | dvds0 11732 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
12 | 1, 11 | ax-mp 5 | . . 3 ⊢ 0 ∥ 0 |
13 | breq2 3980 | . . 3 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0)) | |
14 | 12, 13 | mpbiri 167 | . 2 ⊢ (𝑁 = 0 → 0 ∥ 𝑁) |
15 | 10, 14 | impbid1 141 | 1 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 class class class wbr 3976 (class class class)co 5836 0cc0 7744 · cmul 7749 ℤcz 9182 ∥ cdvds 11713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 df-z 9183 df-dvds 11714 |
This theorem is referenced by: zdvdsdc 11738 dvdsabseq 11770 bezoutlemle 11926 dfgcd3 11928 dfgcd2 11932 dvdssq 11949 rpdvds 12010 pcdvdstr 12237 pc2dvds 12240 |
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