Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > isprm2lem | Unicode version |
Description: Lemma for isprm2 11787. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
isprm2lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 519 | . . . . 5 | |
2 | 1 | necomd 2392 | . . . 4 |
3 | simpr 109 | . . . . 5 | |
4 | nnz 9066 | . . . . . . . 8 | |
5 | 1dvds 11496 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | 6 | ad2antrr 479 | . . . . . 6 |
8 | 1nn 8724 | . . . . . . 7 | |
9 | breq1 3927 | . . . . . . . 8 | |
10 | 9 | elrab3 2836 | . . . . . . 7 |
11 | 8, 10 | ax-mp 5 | . . . . . 6 |
12 | 7, 11 | sylibr 133 | . . . . 5 |
13 | iddvds 11495 | . . . . . . . 8 | |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | 14 | ad2antrr 479 | . . . . . 6 |
16 | breq1 3927 | . . . . . . . 8 | |
17 | 16 | elrab3 2836 | . . . . . . 7 |
18 | 17 | ad2antrr 479 | . . . . . 6 |
19 | 15, 18 | mpbird 166 | . . . . 5 |
20 | en2eqpr 6794 | . . . . 5 | |
21 | 3, 12, 19, 20 | syl3anc 1216 | . . . 4 |
22 | 2, 21 | mpd 13 | . . 3 |
23 | 22 | ex 114 | . 2 |
24 | necom 2390 | . . . 4 | |
25 | pr2ne 7041 | . . . . . 6 | |
26 | 8, 25 | mpan 420 | . . . . 5 |
27 | 26 | biimpar 295 | . . . 4 |
28 | 24, 27 | sylan2br 286 | . . 3 |
29 | breq1 3927 | . . 3 | |
30 | 28, 29 | syl5ibrcom 156 | . 2 |
31 | 23, 30 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2306 crab 2418 cpr 3523 class class class wbr 3924 c2o 6300 cen 6625 c1 7614 cn 8713 cz 9047 cdvds 11482 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1o 6306 df-2o 6307 df-er 6422 df-en 6628 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-z 9048 df-dvds 11483 |
This theorem is referenced by: isprm2 11787 |
Copyright terms: Public domain | W3C validator |