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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 9204 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mul02d 8298 | . 2 ⊢ (𝑁 ∈ ℤ → (0 · 𝑁) = 0) |
3 | 0z 9210 | . . 3 ⊢ 0 ∈ ℤ | |
4 | dvds0lem 11750 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 · 𝑁) = 0) → 𝑁 ∥ 0) | |
5 | 4 | ex 114 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → ((0 · 𝑁) = 0 → 𝑁 ∥ 0)) |
6 | 3, 3, 5 | mp3an13 1323 | . 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 973 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 0cc0 7761 · cmul 7766 ℤcz 9199 ∥ cdvds 11736 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079 df-neg 8080 df-z 9200 df-dvds 11737 |
This theorem is referenced by: 0dvds 11760 alzdvds 11801 fzo0dvdseq 11804 z0even 11857 gcddvds 11905 gcd0id 11921 bezoutlemmain 11940 dfgcd3 11952 dfgcd2 11956 dvdssq 11973 dvdslcm 12010 lcmdvds 12020 mulgcddvds 12035 odzdvds 12186 pcdvdsb 12260 pcz 12272 lgsne0 13654 |
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