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Mirrors > Home > MPE Home > Th. List > dvdsmul1 | Structured version Visualization version Unicode version |
Description: An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmul1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 10949 |
. . 3
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2 | zcn 10949 |
. . 3
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3 | mulcom 9630 |
. . 3
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4 | 1, 2, 3 | syl2anr 481 |
. 2
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5 | zmulcl 10992 |
. . 3
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6 | dvds0lem 14325 |
. . . . 5
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7 | 6 | ex 436 |
. . . 4
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8 | 7 | 3com12 1213 |
. . 3
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9 | 5, 8 | mpd3an3 1367 |
. 2
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10 | 4, 9 | mpd 15 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-er 7368 df-en 7575 df-dom 7576 df-sdom 7577 df-pnf 9682 df-mnf 9683 df-ltxr 9685 df-sub 9867 df-neg 9868 df-nn 10617 df-n0 10877 df-z 10945 df-dvds 14318 |
This theorem is referenced by: dvdsmultr1 14350 ndvdsi 14403 bits0e 14414 bits0o 14415 mulgcd 14526 dvdsmulgcd 14534 lcmcllem 14573 lcmgcdlem 14583 nprm 14650 exprmfct 14660 qredeq 14675 phimullem 14739 prmdiv 14745 opoe 14773 omoe 14774 iserodd 14797 expnprm 14859 pockthlem 14861 prmreclem3 14874 4sqlem14OLD 14914 4sqlem14 14920 odmulg2 17218 odbezout 17221 gexdvds 17247 sylow2alem2 17282 odadd1 17498 odadd2 17499 gexexlem 17502 prmirredlem 19076 znunit 19146 wilthlem2 24006 dvdsflf1o 24128 dvdsmulf1o 24135 ppiublem1 24142 ppiublem2 24143 perfectlem1 24169 bposlem3 24226 lgsdir 24270 lgsquadlem1 24294 lgsquad2lem1 24298 lgsquad2lem2 24299 2sqlem4 24307 2sqblem 24317 dchrisumlem1 24339 ex-ind-dvds 25911 2sqmod 28421 jm2.23 35863 jm2.27c 35874 inductionexd 36605 fouriersw 38105 etransclem24 38133 etransclem28 38137 perfectALTVlem1 38853 |
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