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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 9250 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | divides 11777 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) | |
3 | 1, 2 | mpan 424 | . . 3 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) |
4 | zcn 9244 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
5 | 4 | mul01d 8337 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 0) = 0) |
6 | eqtr2 2196 | . . . . . 6 ⊢ (((𝑛 · 0) = 𝑁 ∧ (𝑛 · 0) = 0) → 𝑁 = 0) | |
7 | 5, 6 | sylan2 286 | . . . . 5 ⊢ (((𝑛 · 0) = 𝑁 ∧ 𝑛 ∈ ℤ) → 𝑁 = 0) |
8 | 7 | ancoms 268 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 · 0) = 𝑁) → 𝑁 = 0) |
9 | 8 | rexlimiva 2589 | . . 3 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁 → 𝑁 = 0) |
10 | 3, 9 | syl6bi 163 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 → 𝑁 = 0)) |
11 | dvds0 11794 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
12 | 1, 11 | ax-mp 5 | . . 3 ⊢ 0 ∥ 0 |
13 | breq2 4004 | . . 3 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0)) | |
14 | 12, 13 | mpbiri 168 | . 2 ⊢ (𝑁 = 0 → 0 ∥ 𝑁) |
15 | 10, 14 | impbid1 142 | 1 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4000 (class class class)co 5869 0cc0 7799 · cmul 7804 ℤcz 9239 ∥ cdvds 11775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-setind 4533 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-sub 8117 df-neg 8118 df-z 9240 df-dvds 11776 |
This theorem is referenced by: zdvdsdc 11800 dvdsabseq 11833 bezoutlemle 11989 dfgcd3 11991 dfgcd2 11995 dvdssq 12012 rpdvds 12079 pcdvdstr 12306 pc2dvds 12309 |
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