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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 | โข (๐ โ โค โ ๐ โฅ ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12505 | . . 3 โข (๐ โ โค โ ๐ โ โ) | |
2 | 1 | mulid2d 11174 | . 2 โข (๐ โ โค โ (1 ยท ๐) = ๐) |
3 | 1z 12534 | . . . 4 โข 1 โ โค | |
4 | dvds0lem 16150 | . . . 4 โข (((1 โ โค โง ๐ โ โค โง ๐ โ โค) โง (1 ยท ๐) = ๐) โ ๐ โฅ ๐) | |
5 | 3, 4 | mp3anl1 1456 | . . 3 โข (((๐ โ โค โง ๐ โ โค) โง (1 ยท ๐) = ๐) โ ๐ โฅ ๐) |
6 | 5 | anabsan 664 | . 2 โข ((๐ โ โค โง (1 ยท ๐) = ๐) โ ๐ โฅ ๐) |
7 | 2, 6 | mpdan 686 | 1 โข (๐ โ โค โ ๐ โฅ ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 class class class wbr 5106 (class class class)co 7358 1c1 11053 ยท cmul 11057 โคcz 12500 โฅ cdvds 16137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rrecex 11124 ax-cnre 11125 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-neg 11389 df-nn 12155 df-z 12501 df-dvds 16138 |
This theorem is referenced by: dvdsadd 16185 dvds1 16202 dvdsext 16204 z2even 16253 divalglem0 16276 divalglem2 16278 sadadd3 16342 gcd0id 16400 gcdzeq 16434 mulgcddvds 16532 1idssfct 16557 isprm2lem 16558 dvdsprime 16564 dvdsprm 16580 exprmfct 16581 coprm 16588 isprm6 16591 pcidlem 16745 pcprmpw2 16755 pcprmpw 16756 prmgaplem1 16922 prmgaplem2 16923 prmgaplcmlem1 16924 prmgaplcmlem2 16925 odeq 19333 pgpfi 19388 znidomb 20971 sgmnncl 26499 muinv 26545 ppiublem2 26554 perfect1 26579 perfectlem2 26581 2sqlem6 26774 ex-ind-dvds 29408 eulerpartlemt 32974 dfgcd3 35798 poimirlem25 36106 poimirlem27 36108 aks4d1p9 40548 jm2.18 41315 jm2.15nn0 41330 jm2.16nn0 41331 jm2.27c 41334 nzss 42604 etransclem25 44507 perfectALTVlem2 45921 |
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