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| Mirrors > Home > ILE Home > Th. List > subgrcl | Unicode version | ||
| Description: Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| Ref | Expression |
|---|---|
| subgrcl |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2204 |
. . 3
| |
| 2 | 1 | issubg 13480 |
. 2
|
| 3 | 2 | simp1bi 1014 |
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-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-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-cnex 8015 ax-resscn 8016 ax-1re 8018 ax-addrcl 8021 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 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-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 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-fv 5278 df-ov 5946 df-inn 9036 df-ndx 12806 df-slot 12807 df-base 12809 df-subg 13477 |
| This theorem is referenced by: subg0 13487 subginv 13488 subgcl 13491 subgsub 13493 subgmulgcl 13494 subgmulg 13495 subgsubm 13503 subsubg 13504 subgintm 13505 isnsg 13509 nsgconj 13513 isnsg3 13514 ssnmz 13518 nmznsg 13520 eqger 13531 eqgid 13533 eqgen 13534 eqgcpbl 13535 qusgrp 13539 quseccl 13540 qusadd 13541 qus0 13542 qusinv 13543 qussub 13544 ecqusaddcl 13546 resghm 13567 resghm2 13568 resghm2b 13569 conjsubg 13584 conjsubgen 13585 conjnmz 13586 conjnmzb 13587 qusghm 13589 issubrng2 13943 issubrg2 13974 |
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