Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqgval | Structured version Visualization version Unicode version |
Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
eqgval.x | |
eqgval.n | |
eqgval.p | |
eqgval.r | ~QG |
Ref | Expression |
---|---|
eqgval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqgval.x | . . . 4 | |
2 | eqgval.n | . . . 4 | |
3 | eqgval.p | . . . 4 | |
4 | eqgval.r | . . . 4 ~QG | |
5 | 1, 2, 3, 4 | eqgfval 17642 | . . 3 |
6 | 5 | breqd 4664 | . 2 |
7 | brabv 6699 | . . . 4 | |
8 | 7 | adantl 482 | . . 3 |
9 | simpr1 1067 | . . . . 5 | |
10 | elex 3212 | . . . . 5 | |
11 | 9, 10 | syl 17 | . . . 4 |
12 | simpr2 1068 | . . . . 5 | |
13 | elex 3212 | . . . . 5 | |
14 | 12, 13 | syl 17 | . . . 4 |
15 | 11, 14 | jca 554 | . . 3 |
16 | vex 3203 | . . . . . . . 8 | |
17 | vex 3203 | . . . . . . . 8 | |
18 | 16, 17 | prss 4351 | . . . . . . 7 |
19 | eleq1 2689 | . . . . . . . 8 | |
20 | eleq1 2689 | . . . . . . . 8 | |
21 | 19, 20 | bi2anan9 917 | . . . . . . 7 |
22 | 18, 21 | syl5bbr 274 | . . . . . 6 |
23 | fveq2 6191 | . . . . . . . 8 | |
24 | id 22 | . . . . . . . 8 | |
25 | 23, 24 | oveqan12d 6669 | . . . . . . 7 |
26 | 25 | eleq1d 2686 | . . . . . 6 |
27 | 22, 26 | anbi12d 747 | . . . . 5 |
28 | df-3an 1039 | . . . . 5 | |
29 | 27, 28 | syl6bbr 278 | . . . 4 |
30 | eqid 2622 | . . . 4 | |
31 | 29, 30 | brabga 4989 | . . 3 |
32 | 8, 15, 31 | pm5.21nd 941 | . 2 |
33 | 6, 32 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cvv 3200 wss 3574 cpr 4179 class class class wbr 4653 copab 4712 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cminusg 17423 ~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-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-rab 2921 df-v 3202 df-sbc 3436 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-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-eqg 17593 |
This theorem is referenced by: eqger 17644 eqglact 17645 eqgid 17646 eqgcpbl 17648 gastacos 17743 orbstafun 17744 sylow2blem1 18035 sylow2blem3 18037 eqgabl 18240 tgpconncompeqg 21915 tgpconncomp 21916 qustgpopn 21923 |
Copyright terms: Public domain | W3C validator |