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Mirrors > Home > ILE Home > Th. List > gcddvds | Unicode version |
Description: The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcddvds |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 8656 |
. . . . . 6
![]() ![]() ![]() ![]() | |
2 | dvds0 10590 |
. . . . . 6
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3 | 1, 2 | ax-mp 7 |
. . . . 5
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4 | breq2 3815 |
. . . . . . 7
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5 | breq2 3815 |
. . . . . . 7
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6 | 4, 5 | bi2anan9 571 |
. . . . . 6
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7 | anidm 388 |
. . . . . 6
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8 | 6, 7 | syl6bb 194 |
. . . . 5
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9 | 3, 8 | mpbiri 166 |
. . . 4
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10 | oveq12 5599 |
. . . . . . 7
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11 | gcd0val 10731 |
. . . . . . 7
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12 | 10, 11 | syl6eq 2131 |
. . . . . 6
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13 | 12 | breq1d 3821 |
. . . . 5
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14 | 12 | breq1d 3821 |
. . . . 5
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15 | 13, 14 | anbi12d 457 |
. . . 4
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16 | 9, 15 | mpbird 165 |
. . 3
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17 | 16 | adantl 271 |
. 2
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18 | gcdn0val 10732 |
. . . 4
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19 | zssre 8652 |
. . . . . 6
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20 | gcdsupex 10728 |
. . . . . 6
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21 | ssrexv 3070 |
. . . . . 6
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22 | 19, 20, 21 | mpsyl 64 |
. . . . 5
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23 | ssrab2 3090 |
. . . . . 6
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24 | 23 | a1i 9 |
. . . . 5
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25 | 22, 24 | suprzclex 8739 |
. . . 4
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26 | 18, 25 | eqeltrd 2159 |
. . 3
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27 | gcdn0cl 10733 |
. . . . 5
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28 | 27 | nnzd 8762 |
. . . 4
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29 | breq1 3814 |
. . . . . 6
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30 | breq1 3814 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | anbi12d 457 |
. . . . 5
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32 | 31 | elrab3 2760 |
. . . 4
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33 | 28, 32 | syl 14 |
. . 3
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34 | 26, 33 | mpbid 145 |
. 2
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35 | gcdmndc 10719 |
. . 3
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36 | exmiddc 778 |
. . 3
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37 | 35, 36 | syl 14 |
. 2
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38 | 17, 34, 37 | mpjaodan 745 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 ax-arch 7366 ax-caucvg 7367 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-recs 6001 df-frec 6087 df-sup 6585 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 df-div 8037 df-inn 8316 df-2 8374 df-3 8375 df-4 8376 df-n0 8565 df-z 8646 df-uz 8914 df-q 8999 df-rp 9029 df-fz 9319 df-fzo 9443 df-fl 9565 df-mod 9618 df-iseq 9740 df-iexp 9791 df-cj 10102 df-re 10103 df-im 10104 df-rsqrt 10257 df-abs 10258 df-dvds 10576 df-gcd 10718 |
This theorem is referenced by: zeqzmulgcd 10741 divgcdz 10742 divgcdnn 10745 gcd0id 10749 gcdneg 10752 gcdaddm 10754 gcd1 10757 dvdsgcdb 10781 dfgcd2 10782 mulgcd 10784 gcdzeq 10790 dvdsmulgcd 10793 sqgcd 10797 dvdssqlem 10798 bezoutr 10800 gcddvdslcm 10834 lcmgcdlem 10838 lcmgcdeq 10844 coprmgcdb 10849 ncoprmgcdne1b 10850 mulgcddvds 10855 rpmulgcd2 10856 qredeu 10858 rpdvds 10860 divgcdcoprm0 10862 divgcdodd 10901 coprm 10902 rpexp 10911 divnumden 10953 phimullem 10980 hashgcdlem 10982 hashgcdeq 10983 |
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