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Mirrors > Home > ILE Home > Th. List > dvds0 | GIF version |
Description: Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds0 | ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 8356 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mul02d 7496 | . 2 ⊢ (𝑁 ∈ ℤ → (0 · 𝑁) = 0) |
3 | 0z 8362 | . . 3 ⊢ 0 ∈ ℤ | |
4 | dvds0lem 10205 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 · 𝑁) = 0) → 𝑁 ∥ 0) | |
5 | 4 | ex 113 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → ((0 · 𝑁) = 0 → 𝑁 ∥ 0)) |
6 | 3, 3, 5 | mp3an13 1259 | . 2 ⊢ (𝑁 ∈ ℤ → ((0 · 𝑁) = 0 → 𝑁 ∥ 0)) |
7 | 2, 6 | mpd 13 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 0cc0 6981 · cmul 6986 ℤcz 8351 ∥ cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 df-z 8352 df-dvds 10196 |
This theorem is referenced by: 0dvds 10215 alzdvds 10254 fzo0dvdseq 10257 z0even 10311 gcddvds 10355 gcd0id 10370 bezoutlemmain 10387 dfgcd3 10399 dfgcd2 10403 dvdssq 10420 dvdslcm 10451 lcmdvds 10461 mulgcddvds 10476 |
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