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| Mirrors > Home > MPE Home > Th. List > gexexlem | Structured version Visualization version Unicode version | ||
| Description: Lemma for gexex 18256. (Contributed by Mario Carneiro, 24-Apr-2016.) |
| Ref | Expression |
|---|---|
| gexex.1 |
        |
| gexex.2 |
 gEx |
| gexex.3 |
  |
| gexexlem.1 |
    |
| gexexlem.2 |
 |
| gexexlem.3 |
 |
| gexexlem.4 |
![/\](wedge.gif "/\"){kind=small}
  |
| Ref | Expression |
|---|---|
| gexexlem |
|
, , , , , ,
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | gexexlem.3 |
. . 3
| |
| 2 | gexex.1 |
. . . 4
| |
| 3 | gexex.3 |
. . . 4
| |
| 4 | 2, 3 | odcl 17955 |
. . 3
 |
| 5 | 1, 4 | syl 17 |
. 2
|
| 6 | gexexlem.2 |
. . 3
| |
| 7 | 6 | nnnn0d 11351 |
. 2
|
| 8 | gexexlem.1 |
. . . 4
| |
| 9 | ablgrp 18198 |
. . . 4
 | |
| 10 | 8, 9 | syl 17 |
. . 3
|
| 11 | gexex.2 |
. . . 4
gEx | |
| 12 | 2, 11, 3 | gexod 18001 |
. . 3
|
| 13 | 10, 1, 12 | syl2anc 693 |
. 2
|
| 14 | 8 | ad2antrr 762 |
. . . . . . . . . . . 12
 |
| 15 | 10 | ad2antrr 762 |
. . . . . . . . . . . . 13
|
| 16 | prmnn 15388 |
. . . . . . . . . . . . . . . 16
| |
| 17 | 16 | adantl 482 |
. . . . . . . . . . . . . . 15
|
| 18 | simpr 477 |
. . . . . . . . . . . . . . . 16
| |
| 19 | 6 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
|
| 20 | 1 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
|
| 21 | 2, 11, 3 | gexnnod 18003 |
. . . . . . . . . . . . . . . . 17
|
| 22 | 15, 19, 20, 21 | syl3anc 1326 |
. . . . . . . . . . . . . . . 16
|
| 23 | 18, 22 | pccld 15555 |
. . . . . . . . . . . . . . 15
|
| 24 | 17, 23 | nnexpcld 13030 |
. . . . . . . . . . . . . 14
 |
| 25 | 24 | nnzd 11481 |
. . . . . . . . . . . . 13
 |
| 26 | eqid 2622 |
. . . . . . . . . . . . . 14
.g .g | |
| 27 | 2, 26 | mulgcl 17559 |
. . . . . . . . . . . . 13
.g |
| 28 | 15, 25, 20, 27 | syl3anc 1326 |
. . . . . . . . . . . 12
.g |
| 29 | simplr 792 |
. . . . . . . . . . . . . . . . 17
| |
| 30 | 2, 11, 3 | gexnnod 18003 |
. . . . . . . . . . . . . . . . 17
|
| 31 | 15, 19, 29, 30 | syl3anc 1326 |
. . . . . . . . . . . . . . . 16
|
| 32 | pcdvds 15568 |
. . . . . . . . . . . . . . . 16
| |
| 33 | 18, 31, 32 | syl2anc 693 |
. . . . . . . . . . . . . . 15
|
| 34 | 18, 31 | pccld 15555 |
. . . . . . . . . . . . . . . . 17
|
| 35 | 17, 34 | nnexpcld 13030 |
. . . . . . . . . . . . . . . 16
|
| 36 | nndivdvds 14989 |
. . . . . . . . . . . . . . . 16
  | |
| 37 | 31, 35, 36 | syl2anc 693 |
. . . . . . . . . . . . . . 15
|
| 38 | 33, 37 | mpbid 222 |
. . . . . . . . . . . . . 14
|
| 39 | 38 | nnzd 11481 |
. . . . . . . . . . . . 13
|
| 40 | 2, 26 | mulgcl 17559 |
. . . . . . . . . . . . 13
.g |
| 41 | 15, 39, 29, 40 | syl3anc 1326 |
. . . . . . . . . . . 12
.g |
| 42 | 2, 3, 26 | odmulg 17973 |
. . . . . . . . . . . . . . . . . 18
 
.g |
| 43 | 15, 20, 25, 42 | syl3anc 1326 |
. . . . . . . . . . . . . . . . 17
.g |
| 44 | pcdvds 15568 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 45 | 18, 22, 44 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
|
| 46 | gcdeq 15272 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 47 | 24, 22, 46 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
|
| 48 | 45, 47 | mpbird 247 |
. . . . . . . . . . . . . . . . . 18
|
| 49 | 48 | oveq1d 6665 |
. . . . . . . . . . . . . . . . 17
.g .g |
| 50 | 43, 49 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
.g |
| 51 | 50 | oveq1d 6665 |
. . . . . . . . . . . . . . 15
.g
|
| 52 | 2, 11, 3 | gexnnod 18003 |
. . . . . . . . . . . . . . . . . 18
.g .g |
| 53 | 15, 19, 28, 52 | syl3anc 1326 |
. . . . . . . . . . . . . . . . 17
.g |
| 54 | 53 | nncnd 11036 |
. . . . . . . . . . . . . . . 16
.g  |
| 55 | 24 | nncnd 11036 |
. . . . . . . . . . . . . . . 16
|
| 56 | 24 | nnne0d 11065 |
. . . . . . . . . . . . . . . 16
  |
| 57 | 54, 55, 56 | divcan3d 10806 |
. . . . . . . . . . . . . . 15
.g
.g |
| 58 | 51, 57 | eqtr2d 2657 |
. . . . . . . . . . . . . 14
.g |
| 59 | 2, 11, 3 | gexnnod 18003 |
. . . . . . . . . . . . . . . . 17
.g
.g |
| 60 | 15, 19, 41, 59 | syl3anc 1326 |
. . . . . . . . . . . . . . . 16
.g |
| 61 | 60 | nncnd 11036 |
. . . . . . . . . . . . . . 15
.g |
| 62 | 35 | nncnd 11036 |
. . . . . . . . . . . . . . 15
|
| 63 | 38 | nncnd 11036 |
. . . . . . . . . . . . . . 15
|
| 64 | 38 | nnne0d 11065 |
. . . . . . . . . . . . . . 15
|
| 65 | 31 | nncnd 11036 |
. . . . . . . . . . . . . . . . 17
|
| 66 | 35 | nnne0d 11065 |
. . . . . . . . . . . . . . . . 17
|
| 67 | 65, 62, 66 | divcan1d 10802 |
. . . . . . . . . . . . . . . 16
|
| 68 | 2, 3, 26 | odmulg 17973 |
. . . . . . . . . . . . . . . . 17
.g |
| 69 | 15, 29, 39, 68 | syl3anc 1326 |
. . . . . . . . . . . . . . . 16
.g |
| 70 | 35 | nnzd 11481 |
. . . . . . . . . . . . . . . . . . . 20
|
| 71 | dvdsmul1 15003 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 72 | 39, 70, 71 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
|
| 73 | 72, 67 | breqtrd 4679 |
. . . . . . . . . . . . . . . . . 18
|
| 74 | gcdeq 15272 |
. . . . . . . . . . . . . . . . . . 19
| |
| 75 | 38, 31, 74 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
|
| 76 | 73, 75 | mpbird 247 |
. . . . . . . . . . . . . . . . 17
|
| 77 | 76 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
.g
.g |
| 78 | 67, 69, 77 | 3eqtrrd 2661 |
. . . . . . . . . . . . . . 15
.g |
| 79 | 61, 62, 63, 64, 78 | mulcanad 10662 |
. . . . . . . . . . . . . 14
.g |
| 80 | 58, 79 | oveq12d 6668 |
. . . . . . . . . . . . 13
.g .g |
| 81 | nndivdvds 14989 |
. . . . . . . . . . . . . . . . 17
| |
| 82 | 22, 24, 81 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
|
| 83 | 45, 82 | mpbid 222 |
. . . . . . . . . . . . . . 15
|
| 84 | 83 | nnzd 11481 |
. . . . . . . . . . . . . 14
|
| 85 | gcdcom 15235 |
. . . . . . . . . . . . . 14
| |
| 86 | 84, 70, 85 | syl2anc 693 |
. . . . . . . . . . . . 13
|
| 87 | pcndvds2 15572 |
. . . . . . . . . . . . . . . 16
| |
| 88 | 18, 22, 87 | syl2anc 693 |
. . . . . . . . . . . . . . 15
|
| 89 | coprm 15423 |
. . . . . . . . . . . . . . . 16
 | |
| 90 | 18, 84, 89 | syl2anc 693 |
. . . . . . . . . . . . . . 15
|
| 91 | 88, 90 | mpbid 222 |
. . . . . . . . . . . . . 14
|
| 92 | prmz 15389 |
. . . . . . . . . . . . . . . 16
| |
| 93 | 92 | adantl 482 |
. . . . . . . . . . . . . . 15
|
| 94 | rpexp1i 15433 |
. . . . . . . . . . . . . . 15
| |
| 95 | 93, 84, 34, 94 | syl3anc 1326 |
. . . . . . . . . . . . . 14
|
| 96 | 91, 95 | mpd 15 |
. . . . . . . . . . . . 13
|
| 97 | 80, 86, 96 | 3eqtrd 2660 |
. . . . . . . . . . . 12
.g .g |
| 98 | eqid 2622 |
. . . . . . . . . . . . 13
 | |
| 99 | 3, 2, 98 | odadd 18253 |
. . . . . . . . . . . 12
.g .g .g
.g
.g .g
.g .g |
| 100 | 14, 28, 41, 97, 99 | syl31anc 1329 |
. . . . . . . . . . 11
.g .g
.g .g |
| 101 | 58, 79 | oveq12d 6668 |
. . . . . . . . . . 11
.g .g |
| 102 | 100, 101 | eqtrd 2656 |
. . . . . . . . . 10
.g .g |
| 103 | 2, 98 | grpcl 17430 |
. . . . . . . . . . . 12
.g .g .g
.g |
| 104 | 15, 28, 41, 103 | syl3anc 1326 |
. . . . . . . . . . 11
.g
.g |
| 105 | gexexlem.4 |
. . . . . . . . . . . . 13
| |
| 106 | 105 | ralrimiva 2966 |
. . . . . . . . . . . 12
 |
| 107 | 106 | ad2antrr 762 |
. . . . . . . . . . 11
|
| 108 | fveq2 6191 |
. . . . . . . . . . . . 13
.g .g
.g .g | |
| 109 | 108 | breq1d 4663 |
. . . . . . . . . . . 12
.g .g
.g
.g |
| 110 | 109 | rspcv 3305 |
. . . . . . . . . . 11
.g
.g .g
.g |
| 111 | 104, 107, 110 | sylc 65 |
. . . . . . . . . 10
.g .g |
| 112 | 102, 111 | eqbrtrrd 4677 |
. . . . . . . . 9
|
| 113 | 83 | nnred 11035 |
. . . . . . . . . 10
 |
| 114 | 22 | nnred 11035 |
. . . . . . . . . 10
|
| 115 | 35 | nnrpd 11870 |
. . . . . . . . . 10
 |
| 116 | 113, 114, 115 | lemuldivd 11921 |
. . . . . . . . 9
|
| 117 | 112, 116 | mpbid 222 |
. . . . . . . 8
|
| 118 | nnrp 11842 |
. . . . . . . . . 10
| |
| 119 | nnrp 11842 |
. . . . . . . . . 10
| |
| 120 | nnrp 11842 |
. . . . . . . . . 10
| |
| 121 | rpregt0 11846 |
. . . . . . . . . . 11
| |
| 122 | rpregt0 11846 |
. . . . . . . . . . 11
| |
| 123 | rpregt0 11846 |
. . . . . . . . . . 11
| |
| 124 | lediv2 10913 |
. . . . . . . . . . 11
| |
| 125 | 121, 122, 123, 124 | syl3an 1368 |
. . . . . . . . . 10
|
| 126 | 118, 119, 120, 125 | syl3an 1368 |
. . . . . . . . 9
|
| 127 | 35, 24, 22, 126 | syl3anc 1326 |
. . . . . . . 8
|
| 128 | 117, 127 | mpbird 247 |
. . . . . . 7
|
| 129 | 17 | nnred 11035 |
. . . . . . . 8
|
| 130 | 34 | nn0zd 11480 |
. . . . . . . 8
|
| 131 | 23 | nn0zd 11480 |
. . . . . . . 8
|
| 132 | prmuz2 15408 |
. . . . . . . . . 10
  | |
| 133 | 132 | adantl 482 |
. . . . . . . . 9
|
| 134 | eluz2b2 11761 |
. . . . . . . . . 10
| |
| 135 | 134 | simprbi 480 |
. . . . . . . . 9
|
| 136 | 133, 135 | syl 17 |
. . . . . . . 8
|
| 137 | 129, 130, 131, 136 | leexp2d 13039 |
. . . . . . 7
|
| 138 | 128, 137 | mpbird 247 |
. . . . . 6
|
| 139 | 138 | ralrimiva 2966 |
. . . . 5
|
| 140 | 2, 3 | odcl 17955 |
. . . . . . . 8
|
| 141 | 140 | adantl 482 |
. . . . . . 7
|
| 142 | 141 | nn0zd 11480 |
. . . . . 6
|
| 143 | 5 | nn0zd 11480 |
. . . . . . 7
|
| 144 | 143 | adantr 481 |
. . . . . 6
|
| 145 | pc2dvds 15583 |
. . . . . 6
| |
| 146 | 142, 144, 145 | syl2anc 693 |
. . . . 5
|
| 147 | 139, 146 | mpbird 247 |
. . . 4
|
| 148 | 147 | ralrimiva 2966 |
. . 3
|
| 149 | 2, 11, 3 | gexdvds2 18000 |
. . . 4
|
| 150 | 10, 143, 149 | syl2anc 693 |
. . 3
|
| 151 | 148, 150 | mpbird 247 |
. 2
|
| 152 | dvdseq 15036 |
. 2
| |
| 153 | 5, 7, 13, 151, 152 | syl22anc 1327 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 class class class wbr 4653
cfv 5888
(class class class)co 6650 cr 9935 cc0 9936 c1 9937 cmul 9941 clt 10074 cle 10075 cdiv 10684 cn 11020 c2 11070 cn0 11292 cz 11377 cuz 11687 crp 11832 cexp 12860 cdvds 14983 cgcd 15216 cprime 15385 cpc 15541 cbs 15857 cplusg 15941 cgrp 17422 .gcmg 17540
cod 17944
gExcgex 17945 cabl 18194 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-od 17948 df-gex 17949 df-cmn 18195 df-abl 18196 |
| This theorem is referenced by: gexex 18256 |
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