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Mirrors > Home > ILE Home > Th. List > dvds0lem | Unicode version |
Description: A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds0lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5833 | . . . . . . . . 9 | |
2 | 1 | eqeq1d 2166 | . . . . . . . 8 |
3 | 2 | rspcev 2816 | . . . . . . 7 |
4 | 3 | adantl 275 | . . . . . 6 |
5 | divides 11696 | . . . . . . 7 | |
6 | 5 | adantr 274 | . . . . . 6 |
7 | 4, 6 | mpbird 166 | . . . . 5 |
8 | 7 | expr 373 | . . . 4 |
9 | 8 | 3impa 1177 | . . 3 |
10 | 9 | 3comr 1193 | . 2 |
11 | 10 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wrex 2436 class class class wbr 3967 (class class class)co 5826 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-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 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 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-rex 2441 df-v 2714 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-br 3968 df-opab 4028 df-iota 5137 df-fv 5180 df-ov 5829 df-dvds 11695 |
This theorem is referenced by: iddvds 11711 1dvds 11712 dvds0 11713 dvdsmul1 11720 dvdsmul2 11721 divalgmod 11830 oddpwdclemxy 12059 ex-dvds 13377 |
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