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Mirrors > Home > MPE Home > Th. List > dprdsn | Structured version Visualization version Unicode version |
Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdsn | SubGrp DProd DProd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 Cntz Cntz | |
2 | eqid 2622 | . . 3 | |
3 | eqid 2622 | . . 3 mrClsSubGrp mrClsSubGrp | |
4 | subgrcl 17599 | . . . 4 SubGrp | |
5 | 4 | adantl 482 | . . 3 SubGrp |
6 | snex 4908 | . . . 4 | |
7 | 6 | a1i 11 | . . 3 SubGrp |
8 | f1osng 6177 | . . . . 5 SubGrp | |
9 | f1of 6137 | . . . . 5 | |
10 | 8, 9 | syl 17 | . . . 4 SubGrp |
11 | simpr 477 | . . . . 5 SubGrp SubGrp | |
12 | 11 | snssd 4340 | . . . 4 SubGrp SubGrp |
13 | 10, 12 | fssd 6057 | . . 3 SubGrp SubGrp |
14 | simpr1 1067 | . . . . . 6 SubGrp | |
15 | elsni 4194 | . . . . . 6 | |
16 | 14, 15 | syl 17 | . . . . 5 SubGrp |
17 | simpr2 1068 | . . . . . 6 SubGrp | |
18 | elsni 4194 | . . . . . 6 | |
19 | 17, 18 | syl 17 | . . . . 5 SubGrp |
20 | 16, 19 | eqtr4d 2659 | . . . 4 SubGrp |
21 | simpr3 1069 | . . . 4 SubGrp | |
22 | 20, 21 | pm2.21ddne 2878 | . . 3 SubGrp Cntz |
23 | 5 | adantr 481 | . . . . . . . 8 SubGrp |
24 | eqid 2622 | . . . . . . . . 9 | |
25 | 24 | subgacs 17629 | . . . . . . . 8 SubGrp ACS |
26 | acsmre 16313 | . . . . . . . 8 SubGrp ACS SubGrp Moore | |
27 | 23, 25, 26 | 3syl 18 | . . . . . . 7 SubGrp SubGrp Moore |
28 | 15 | adantl 482 | . . . . . . . . . . . . . . 15 SubGrp |
29 | 28 | sneqd 4189 | . . . . . . . . . . . . . 14 SubGrp |
30 | 29 | difeq2d 3728 | . . . . . . . . . . . . 13 SubGrp |
31 | difid 3948 | . . . . . . . . . . . . 13 | |
32 | 30, 31 | syl6eq 2672 | . . . . . . . . . . . 12 SubGrp |
33 | 32 | imaeq2d 5466 | . . . . . . . . . . 11 SubGrp |
34 | ima0 5481 | . . . . . . . . . . 11 | |
35 | 33, 34 | syl6eq 2672 | . . . . . . . . . 10 SubGrp |
36 | 35 | unieqd 4446 | . . . . . . . . 9 SubGrp |
37 | uni0 4465 | . . . . . . . . 9 | |
38 | 36, 37 | syl6eq 2672 | . . . . . . . 8 SubGrp |
39 | 0ss 3972 | . . . . . . . . 9 | |
40 | 39 | a1i 11 | . . . . . . . 8 SubGrp |
41 | 38, 40 | eqsstrd 3639 | . . . . . . 7 SubGrp |
42 | 2 | 0subg 17619 | . . . . . . . 8 SubGrp |
43 | 23, 42 | syl 17 | . . . . . . 7 SubGrp SubGrp |
44 | 3 | mrcsscl 16280 | . . . . . . 7 SubGrp Moore SubGrp mrClsSubGrp |
45 | 27, 41, 43, 44 | syl3anc 1326 | . . . . . 6 SubGrp mrClsSubGrp |
46 | 2 | subg0cl 17602 | . . . . . . . . 9 SubGrp |
47 | 46 | ad2antlr 763 | . . . . . . . 8 SubGrp |
48 | 15 | fveq2d 6195 | . . . . . . . . 9 |
49 | fvsng 6447 | . . . . . . . . 9 SubGrp | |
50 | 48, 49 | sylan9eqr 2678 | . . . . . . . 8 SubGrp |
51 | 47, 50 | eleqtrrd 2704 | . . . . . . 7 SubGrp |
52 | 51 | snssd 4340 | . . . . . 6 SubGrp |
53 | 45, 52 | sstrd 3613 | . . . . 5 SubGrp mrClsSubGrp |
54 | sseqin2 3817 | . . . . 5 mrClsSubGrp mrClsSubGrp mrClsSubGrp | |
55 | 53, 54 | sylib 208 | . . . 4 SubGrp mrClsSubGrp mrClsSubGrp |
56 | 55, 45 | eqsstrd 3639 | . . 3 SubGrp mrClsSubGrp |
57 | 1, 2, 3, 5, 7, 13, 22, 56 | dmdprdd 18398 | . 2 SubGrp DProd |
58 | 3 | dprdspan 18426 | . . . 4 DProd DProd mrClsSubGrp |
59 | 57, 58 | syl 17 | . . 3 SubGrp DProd mrClsSubGrp |
60 | rnsnopg 5614 | . . . . . . . 8 | |
61 | 60 | adantr 481 | . . . . . . 7 SubGrp |
62 | 61 | unieqd 4446 | . . . . . 6 SubGrp |
63 | unisng 4452 | . . . . . . 7 SubGrp | |
64 | 63 | adantl 482 | . . . . . 6 SubGrp |
65 | 62, 64 | eqtrd 2656 | . . . . 5 SubGrp |
66 | 65 | fveq2d 6195 | . . . 4 SubGrp mrClsSubGrp mrClsSubGrp |
67 | 5, 25, 26 | 3syl 18 | . . . . 5 SubGrp SubGrp Moore |
68 | 3 | mrcid 16273 | . . . . 5 SubGrp Moore SubGrp mrClsSubGrp |
69 | 67, 68 | sylancom 701 | . . . 4 SubGrp mrClsSubGrp |
70 | 66, 69 | eqtrd 2656 | . . 3 SubGrp mrClsSubGrp |
71 | 59, 70 | eqtrd 2656 | . 2 SubGrp DProd |
72 | 57, 71 | jca 554 | 1 SubGrp DProd DProd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 cvv 3200 cdif 3571 cin 3573 wss 3574 c0 3915 csn 4177 cop 4183 cuni 4436 class class class wbr 4653 cdm 5114 crn 5115 cima 5117 wf 5884 wf1o 5887 cfv 5888 (class class class)co 6650 cbs 15857 c0g 16100 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245 cgrp 17422 SubGrpcsubg 17588 Cntzccntz 17748 DProd cdprd 18392 |
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 |
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-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-cmn 18195 df-dprd 18394 |
This theorem is referenced by: dprd2da 18441 dmdprdpr 18448 dprdpr 18449 dpjlem 18450 pgpfaclem1 18480 |
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