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Mirrors > Home > ILE Home > Th. List > isprm2lem | Unicode version |
Description: Lemma for isprm2 10499. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
isprm2lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 496 | . . . . 5 | |
2 | 1 | necomd 2331 | . . . 4 |
3 | simpr 108 | . . . . 5 | |
4 | nnz 8370 | . . . . . . . 8 | |
5 | 1dvds 10209 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | 6 | ad2antrr 471 | . . . . . 6 |
8 | 1nn 8050 | . . . . . . 7 | |
9 | breq1 3788 | . . . . . . . 8 | |
10 | 9 | elrab3 2750 | . . . . . . 7 |
11 | 8, 10 | ax-mp 7 | . . . . . 6 |
12 | 7, 11 | sylibr 132 | . . . . 5 |
13 | iddvds 10208 | . . . . . . . 8 | |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | 14 | ad2antrr 471 | . . . . . 6 |
16 | breq1 3788 | . . . . . . . 8 | |
17 | 16 | elrab3 2750 | . . . . . . 7 |
18 | 17 | ad2antrr 471 | . . . . . 6 |
19 | 15, 18 | mpbird 165 | . . . . 5 |
20 | en2eqpr 6380 | . . . . 5 | |
21 | 3, 12, 19, 20 | syl3anc 1169 | . . . 4 |
22 | 2, 21 | mpd 13 | . . 3 |
23 | 22 | ex 113 | . 2 |
24 | necom 2329 | . . . 4 | |
25 | pr2ne 6461 | . . . . . 6 | |
26 | 8, 25 | mpan 414 | . . . . 5 |
27 | 26 | biimpar 291 | . . . 4 |
28 | 24, 27 | sylan2br 282 | . . 3 |
29 | breq1 3788 | . . 3 | |
30 | 28, 29 | syl5ibrcom 155 | . 2 |
31 | 23, 30 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wne 2245 crab 2352 cpr 3399 class class class wbr 3785 c2o 6018 cen 6242 c1 6982 cn 8039 cz 8351 cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1o 6024 df-2o 6025 df-er 6129 df-en 6245 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-z 8352 df-dvds 10196 |
This theorem is referenced by: isprm2 10499 |
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