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| Mirrors > Home > ILE Home > Th. List > coprm | Unicode version | ||
| Description: A prime number either divides an integer or is coprime to it, but not both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 22-Jun-2011.) |
| Ref | Expression |
|---|---|
| coprm |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prmz 12404 |
. . . . . . 7
| |
| 2 | gcddvds 12255 |
. . . . . . 7
| |
| 3 | 1, 2 | sylan 283 |
. . . . . 6
|
| 4 | 3 | simprd 114 |
. . . . 5
|
| 5 | breq1 4046 |
. . . . 5
| |
| 6 | 4, 5 | syl5ibcom 155 |
. . . 4
|
| 7 | 6 | con3d 632 |
. . 3
|
| 8 | 0nnn 9062 |
. . . . . . . . 9
| |
| 9 | prmnn 12403 |
. . . . . . . . . 10
| |
| 10 | eleq1 2267 |
. . . . . . . . . 10
| |
| 11 | 9, 10 | syl5ibcom 155 |
. . . . . . . . 9
|
| 12 | 8, 11 | mtoi 665 |
. . . . . . . 8
|
| 13 | 12 | intnanrd 933 |
. . . . . . 7
|
| 14 | 13 | adantr 276 |
. . . . . 6
|
| 15 | gcdn0cl 12254 |
. . . . . . . 8
| |
| 16 | 15 | ex 115 |
. . . . . . 7
|
| 17 | 1, 16 | sylan 283 |
. . . . . 6
|
| 18 | 14, 17 | mpd 13 |
. . . . 5
|
| 19 | 3 | simpld 112 |
. . . . 5
|
| 20 | isprm2 12410 |
. . . . . . . 8
| |
| 21 | 20 | simprbi 275 |
. . . . . . 7
|
| 22 | breq1 4046 |
. . . . . . . . 9
| |
| 23 | eqeq1 2211 |
. . . . . . . . . 10
| |
| 24 | eqeq1 2211 |
. . . . . . . . . 10
| |
| 25 | 23, 24 | orbi12d 794 |
. . . . . . . . 9
|
| 26 | 22, 25 | imbi12d 234 |
. . . . . . . 8
|
| 27 | 26 | rspcv 2872 |
. . . . . . 7
|
| 28 | 21, 27 | syl5com 29 |
. . . . . 6
|
| 29 | 28 | adantr 276 |
. . . . 5
|
| 30 | 18, 19, 29 | mp2d 47 |
. . . 4
|
| 31 | biorf 745 |
. . . . 5
| |
| 32 | orcom 729 |
. . . . 5
| |
| 33 | 31, 32 | bitrdi 196 |
. . . 4
|
| 34 | 30, 33 | syl5ibrcom 157 |
. . 3
|
| 35 | 7, 34 | syld 45 |
. 2
|
| 36 | iddvds 12086 |
. . . . . . 7
| |
| 37 | 1, 36 | syl 14 |
. . . . . 6
|
| 38 | 37 | adantr 276 |
. . . . 5
|
| 39 | dvdslegcd 12256 |
. . . . . . . . 9
| |
| 40 | 39 | ex 115 |
. . . . . . . 8
|
| 41 | 40 | 3anidm12 1307 |
. . . . . . 7
|
| 42 | 1, 41 | sylan 283 |
. . . . . 6
|
| 43 | 14, 42 | mpd 13 |
. . . . 5
|
| 44 | 38, 43 | mpand 429 |
. . . 4
|
| 45 | prmgt1 12425 |
. . . . . 6
| |
| 46 | 45 | adantr 276 |
. . . . 5
|
| 47 | 1 | zred 9494 |
. . . . . . 7
|
| 48 | 47 | adantr 276 |
. . . . . 6
|
| 49 | 18 | nnred 9048 |
. . . . . 6
|
| 50 | 1re 8070 |
. . . . . . 7
| |
| 51 | ltletr 8161 |
. . . . . . 7
| |
| 52 | 50, 51 | mp3an1 1336 |
. . . . . 6
|
| 53 | 48, 49, 52 | syl2anc 411 |
. . . . 5
|
| 54 | 46, 53 | mpand 429 |
. . . 4
|
| 55 | ltne 8156 |
. . . . . 6
| |
| 56 | 50, 55 | mpan 424 |
. . . . 5
|
| 57 | 56 | a1i 9 |
. . . 4
|
| 58 | 44, 54, 57 | 3syld 57 |
. . 3
|
| 59 | 58 | necon2bd 2433 |
. 2
|
| 60 | 35, 59 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-frec 6476 df-1o 6501 df-2o 6502 df-er 6619 df-en 6827 df-sup 7085 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-fz 10130 df-fzo 10264 df-fl 10411 df-mod 10466 df-seqfrec 10591 df-exp 10682 df-cj 11124 df-re 11125 df-im 11126 df-rsqrt 11280 df-abs 11281 df-dvds 12070 df-gcd 12246 df-prm 12401 |
| This theorem is referenced by: prmrp 12438 euclemma 12439 cncongrprm 12450 isoddgcd1 12452 phiprmpw 12515 fermltl 12527 prmdiv 12528 prmdiveq 12529 vfermltl 12545 prmpwdvds 12649 perfect1 15441 perfectlem1 15442 perfectlem2 15443 lgslem1 15448 lgsprme0 15490 gausslemma2dlem0c 15499 lgseisenlem3 15520 lgsquad2lem2 15530 |
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