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Mirrors > Home > ILE Home > Th. List > dvds2lem | Unicode version |
Description: A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds2lem.1 | |
dvds2lem.2 | |
dvds2lem.3 | |
dvds2lem.4 | |
dvds2lem.5 |
Ref | Expression |
---|---|
dvds2lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvds2lem.1 | . . . . . 6 | |
2 | dvds2lem.2 | . . . . . 6 | |
3 | divides 11422 | . . . . . . 7 | |
4 | divides 11422 | . . . . . . 7 | |
5 | 3, 4 | bi2anan9 580 | . . . . . 6 |
6 | 1, 2, 5 | syl2anc 408 | . . . . 5 |
7 | 6 | biimpd 143 | . . . 4 |
8 | reeanv 2577 | . . . 4 | |
9 | 7, 8 | syl6ibr 161 | . . 3 |
10 | dvds2lem.4 | . . . . 5 | |
11 | dvds2lem.5 | . . . . 5 | |
12 | oveq1 5749 | . . . . . . 7 | |
13 | 12 | eqeq1d 2126 | . . . . . 6 |
14 | 13 | rspcev 2763 | . . . . 5 |
15 | 10, 11, 14 | syl6an 1395 | . . . 4 |
16 | 15 | rexlimdvva 2534 | . . 3 |
17 | 9, 16 | syld 45 | . 2 |
18 | dvds2lem.3 | . . 3 | |
19 | divides 11422 | . . 3 | |
20 | 18, 19 | syl 14 | . 2 |
21 | 17, 20 | sylibrd 168 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wrex 2394 class class class wbr 3899 (class class class)co 5742 cmul 7593 cz 9022 cdvds 11420 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-iota 5058 df-fv 5101 df-ov 5745 df-dvds 11421 |
This theorem is referenced by: dvds2ln 11453 dvds2add 11454 dvds2sub 11455 dvdstr 11457 |
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