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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 8853 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mul02d 7967 | . 2 ⊢ (𝑁 ∈ ℤ → (0 · 𝑁) = 0) |
3 | 0z 8859 | . . 3 ⊢ 0 ∈ ℤ | |
4 | dvds0lem 11233 | . . . 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 1271 | . 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 927 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 0cc0 7447 · cmul 7452 ℤcz 8848 ∥ cdvds 11223 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-sub 7752 df-neg 7753 df-z 8849 df-dvds 11224 |
This theorem is referenced by: 0dvds 11243 alzdvds 11282 fzo0dvdseq 11285 z0even 11338 gcddvds 11382 gcd0id 11397 bezoutlemmain 11414 dfgcd3 11426 dfgcd2 11430 dvdssq 11447 dvdslcm 11478 lcmdvds 11488 mulgcddvds 11503 |
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