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Mirrors > Home > MPE Home > Th. List > eqgen | Structured version Visualization version Unicode version |
Description: Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.) |
Ref | Expression |
---|---|
eqger.x | |
eqger.r | ~QG |
Ref | Expression |
---|---|
eqgen | SubGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . 2 | |
2 | breq2 4657 | . 2 | |
3 | simpl 473 | . . . 4 SubGrp SubGrp | |
4 | subgrcl 17599 | . . . . . . 7 SubGrp | |
5 | eqger.x | . . . . . . . 8 | |
6 | 5 | subgss 17595 | . . . . . . 7 SubGrp |
7 | 4, 6 | jca 554 | . . . . . 6 SubGrp |
8 | eqger.r | . . . . . . . 8 ~QG | |
9 | eqid 2622 | . . . . . . . 8 | |
10 | 5, 8, 9 | eqglact 17645 | . . . . . . 7 |
11 | 10 | 3expa 1265 | . . . . . 6 |
12 | 7, 11 | sylan 488 | . . . . 5 SubGrp |
13 | ovex 6678 | . . . . . . 7 ~QG | |
14 | 8, 13 | eqeltri 2697 | . . . . . 6 |
15 | ecexg 7746 | . . . . . 6 | |
16 | 14, 15 | ax-mp 5 | . . . . 5 |
17 | 12, 16 | syl6eqelr 2710 | . . . 4 SubGrp |
18 | eqid 2622 | . . . . . . . . 9 | |
19 | 18, 5, 9 | grplactf1o 17519 | . . . . . . . 8 |
20 | 18, 5 | grplactfval 17516 | . . . . . . . . . 10 |
21 | 20 | adantl 482 | . . . . . . . . 9 |
22 | f1oeq1 6127 | . . . . . . . . 9 | |
23 | 21, 22 | syl 17 | . . . . . . . 8 |
24 | 19, 23 | mpbid 222 | . . . . . . 7 |
25 | 4, 24 | sylan 488 | . . . . . 6 SubGrp |
26 | f1of1 6136 | . . . . . 6 | |
27 | 25, 26 | syl 17 | . . . . 5 SubGrp |
28 | 6 | adantr 481 | . . . . 5 SubGrp |
29 | f1ores 6151 | . . . . 5 | |
30 | 27, 28, 29 | syl2anc 693 | . . . 4 SubGrp |
31 | f1oen2g 7972 | . . . 4 SubGrp | |
32 | 3, 17, 30, 31 | syl3anc 1326 | . . 3 SubGrp |
33 | 32, 12 | breqtrrd 4681 | . 2 SubGrp |
34 | 1, 2, 33 | ectocld 7814 | 1 SubGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cvv 3200 wss 3574 class class class wbr 4653 cmpt 4729 cres 5116 cima 5117 wf1 5885 wf1o 5887 cfv 5888 (class class class)co 6650 cec 7740 cqs 7741 cen 7952 cbs 15857 cplusg 15941 cgrp 17422 SubGrpcsubg 17588 ~QG cqg 17590 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-ec 7744 df-qs 7748 df-en 7956 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-eqg 17593 |
This theorem is referenced by: lagsubg2 17655 sylow2blem1 18035 |
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