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Mirrors > Home > ILE Home > Th. List > dvds1lem | Unicode version |
Description: A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds1lem.1 | |
dvds1lem.2 | |
dvds1lem.3 | |
dvds1lem.4 |
Ref | Expression |
---|---|
dvds1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvds1lem.3 | . . . 4 | |
2 | dvds1lem.4 | . . . 4 | |
3 | oveq1 5843 | . . . . . 6 | |
4 | 3 | eqeq1d 2173 | . . . . 5 |
5 | 4 | rspcev 2825 | . . . 4 |
6 | 1, 2, 5 | syl6an 1421 | . . 3 |
7 | 6 | rexlimdva 2581 | . 2 |
8 | dvds1lem.1 | . . 3 | |
9 | divides 11715 | . . 3 | |
10 | 8, 9 | syl 14 | . 2 |
11 | dvds1lem.2 | . . 3 | |
12 | divides 11715 | . . 3 | |
13 | 11, 12 | syl 14 | . 2 |
14 | 7, 10, 13 | 3imtr4d 202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cmul 7749 cz 9182 cdvds 11713 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-iota 5147 df-fv 5190 df-ov 5839 df-dvds 11714 |
This theorem is referenced by: negdvdsb 11733 dvdsnegb 11734 muldvds1 11742 muldvds2 11743 dvdscmul 11744 dvdsmulc 11745 dvdscmulr 11746 dvdsmulcr 11747 |
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