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Mirrors > Home > ILE Home > Th. List > muldvds1 | Unicode version |
Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
muldvds1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zmulcl 9225 | . . . 4 | |
2 | 1 | anim1i 338 | . . 3 |
3 | 2 | 3impa 1177 | . 2 |
4 | 3simpb 980 | . 2 | |
5 | zmulcl 9225 | . . . 4 | |
6 | 5 | ancoms 266 | . . 3 |
7 | 6 | 3ad2antl2 1145 | . 2 |
8 | zcn 9177 | . . . . . . . 8 | |
9 | zcn 9177 | . . . . . . . 8 | |
10 | zcn 9177 | . . . . . . . 8 | |
11 | mulass 7865 | . . . . . . . . 9 | |
12 | mul32 8009 | . . . . . . . . 9 | |
13 | 11, 12 | eqtr3d 2192 | . . . . . . . 8 |
14 | 8, 9, 10, 13 | syl3an 1262 | . . . . . . 7 |
15 | 14 | 3coml 1192 | . . . . . 6 |
16 | 15 | 3expa 1185 | . . . . 5 |
17 | 16 | 3adantl3 1140 | . . . 4 |
18 | 17 | eqeq1d 2166 | . . 3 |
19 | 18 | biimpd 143 | . 2 |
20 | 3, 4, 7, 19 | dvds1lem 11709 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 class class class wbr 3967 (class class class)co 5826 cc 7732 cmul 7739 cz 9172 cdvds 11694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-sub 8052 df-neg 8053 df-inn 8839 df-n0 9096 df-z 9173 df-dvds 11695 |
This theorem is referenced by: pw2dvdseulemle 12057 |
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