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Mirrors > Home > MPE Home > Th. List > 0dvds | Structured version Visualization version 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 11426 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | divides 15029 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) | |
3 | 1, 2 | mpan 706 | . . 3 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁)) |
4 | zcn 11420 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
5 | 4 | mul01d 10273 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 0) = 0) |
6 | eqtr2 2671 | . . . . . 6 ⊢ (((𝑛 · 0) = 𝑁 ∧ (𝑛 · 0) = 0) → 𝑁 = 0) | |
7 | 5, 6 | sylan2 490 | . . . . 5 ⊢ (((𝑛 · 0) = 𝑁 ∧ 𝑛 ∈ ℤ) → 𝑁 = 0) |
8 | 7 | ancoms 468 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 · 0) = 𝑁) → 𝑁 = 0) |
9 | 8 | rexlimiva 3057 | . . 3 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 0) = 𝑁 → 𝑁 = 0) |
10 | 3, 9 | syl6bi 243 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 → 𝑁 = 0)) |
11 | dvds0 15044 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
12 | 1, 11 | ax-mp 5 | . . 3 ⊢ 0 ∥ 0 |
13 | breq2 4689 | . . 3 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0)) | |
14 | 12, 13 | mpbiri 248 | . 2 ⊢ (𝑁 = 0 → 0 ∥ 𝑁) |
15 | 10, 14 | impbid1 215 | 1 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∈ wcel 2030 ∃wrex 2942 class class class wbr 4685 (class class class)co 6690 0cc0 9974 · cmul 9979 ℤcz 11415 ∥ cdvds 15027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-neg 10307 df-z 11416 df-dvds 15028 |
This theorem is referenced by: fsumdvds 15077 dvdsabseq 15082 dfgcd2 15310 dvdssq 15327 rpdvds 15421 pcdvdstr 15627 pc2dvds 15630 mndodcongi 18008 oddvdsnn0 18009 oddvds 18012 odmulgeq 18020 odf1 18025 odf1o1 18033 gexdvds 18045 gexnnod 18049 torsubg 18303 znf1o 19948 jm2.19 37877 nzss 38833 |
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