![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iddvds | Structured version Visualization version GIF version |
Description: An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
iddvds | ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12462 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | mulid2d 11131 | . 2 ⊢ (𝑁 ∈ ℤ → (1 · 𝑁) = 𝑁) |
3 | 1z 12491 | . . . 4 ⊢ 1 ∈ ℤ | |
4 | dvds0lem 16109 | . . . 4 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) | |
5 | 3, 4 | mp3anl1 1455 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) |
6 | 5 | anabsan 663 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (1 · 𝑁) = 𝑁) → 𝑁 ∥ 𝑁) |
7 | 2, 6 | mpdan 685 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7351 1c1 11010 · cmul 11014 ℤcz 12457 ∥ cdvds 16096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rrecex 11081 ax-cnre 11082 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-neg 11346 df-nn 12112 df-z 12458 df-dvds 16097 |
This theorem is referenced by: dvdsadd 16144 dvds1 16161 dvdsext 16163 z2even 16212 divalglem0 16235 divalglem2 16237 sadadd3 16301 gcd0id 16359 gcdzeq 16393 mulgcddvds 16491 1idssfct 16516 isprm2lem 16517 dvdsprime 16523 dvdsprm 16539 exprmfct 16540 coprm 16547 isprm6 16550 pcidlem 16704 pcprmpw2 16714 pcprmpw 16715 prmgaplem1 16881 prmgaplem2 16882 prmgaplcmlem1 16883 prmgaplcmlem2 16884 odeq 19291 pgpfi 19346 znidomb 20921 sgmnncl 26448 muinv 26494 ppiublem2 26503 perfect1 26528 perfectlem2 26530 2sqlem6 26723 ex-ind-dvds 29234 eulerpartlemt 32783 dfgcd3 35733 poimirlem25 36041 poimirlem27 36043 aks4d1p9 40483 jm2.18 41221 jm2.15nn0 41236 jm2.16nn0 41237 jm2.27c 41240 nzss 42508 etransclem25 44401 perfectALTVlem2 45815 |
Copyright terms: Public domain | W3C validator |