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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 11840 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mul02d 10691 | . 2 ⊢ (𝑁 ∈ ℤ → (0 · 𝑁) = 0) |
3 | 0z 11846 | . . 3 ⊢ 0 ∈ ℤ | |
4 | dvds0lem 15457 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 · 𝑁) = 0) → 𝑁 ∥ 0) | |
5 | 4 | ex 413 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → ((0 · 𝑁) = 0 → 𝑁 ∥ 0)) |
6 | 3, 3, 5 | mp3an13 1444 | . 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 1080 = wceq 1525 ∈ wcel 2083 class class class wbr 4968 (class class class)co 7023 0cc0 10390 · cmul 10395 ℤcz 11835 ∥ cdvds 15444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-ov 7026 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-ltxr 10533 df-neg 10726 df-z 11836 df-dvds 15445 |
This theorem is referenced by: 0dvds 15467 fsumdvds 15495 alzdvds 15507 fzo0dvdseq 15510 z0even 15553 sadadd3 15647 gcddvds 15689 gcd0id 15704 bezoutlem4 15723 dfgcd2 15727 dvdssq 15744 dvdslcm 15775 lcmdvds 15785 dvdslcmf 15808 mulgcddvds 15832 odzdvds 15965 pcdvdsb 16038 pcz 16050 sylow2blem3 18481 odadd1 18695 odadd2 18696 cyggex2 18742 lgsne0 25597 lgsqr 25613 nn0prpw 33282 poimirlem25 34469 poimirlem26 34470 poimirlem27 34471 poimirlem28 34472 congid 39074 jm2.18 39091 jm2.19 39096 jm2.22 39098 jm2.23 39099 etransclem24 42107 etransclem25 42108 etransclem28 42111 |
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