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Mirrors > Home > MPE Home > Th. List > dvds0 | Structured version Visualization version 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 11420 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mul02d 10272 | . 2 ⊢ (𝑁 ∈ ℤ → (0 · 𝑁) = 0) |
3 | 0z 11426 | . . 3 ⊢ 0 ∈ ℤ | |
4 | dvds0lem 15039 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 · 𝑁) = 0) → 𝑁 ∥ 0) | |
5 | 4 | ex 449 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → ((0 · 𝑁) = 0 → 𝑁 ∥ 0)) |
6 | 3, 3, 5 | mp3an13 1455 | . 2 ⊢ (𝑁 ∈ ℤ → ((0 · 𝑁) = 0 → 𝑁 ∥ 0)) |
7 | 2, 6 | mpd 15 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 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: 0dvds 15049 fsumdvds 15077 alzdvds 15089 fzo0dvdseq 15092 z0even 15150 bitsfzo 15204 bitsmod 15205 bitsinv1lem 15210 sadadd3 15230 gcddvds 15272 gcd0id 15287 bezoutlem4 15306 dfgcd2 15310 dvdssq 15327 dvdslcm 15358 lcmdvds 15368 dvdslcmf 15391 mulgcddvds 15416 odzdvds 15547 pcdvdsb 15620 pcz 15632 sylow2blem3 18083 odadd1 18297 odadd2 18298 cyggex2 18344 ppiublem2 24973 lgsdir2lem3 25097 lgsne0 25105 lgsqr 25121 eupth2lem3lem3 27208 eupth2lemb 27215 nn0prpw 32443 poimirlem25 33564 poimirlem26 33565 poimirlem27 33566 poimirlem28 33567 congid 37855 jm2.18 37872 jm2.19 37877 jm2.22 37879 jm2.23 37880 etransclem24 40793 etransclem25 40794 etransclem28 40797 |
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