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Mirrors > Home > MPE Home > Th. List > subsubg | Structured version Visualization version Unicode version |
Description: A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.) |
Ref | Expression |
---|---|
subsubg.h | ↾s |
Ref | Expression |
---|---|
subsubg | SubGrp SubGrp SubGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 17599 | . . . . 5 SubGrp | |
2 | 1 | adantr 481 | . . . 4 SubGrp SubGrp |
3 | eqid 2622 | . . . . . . . 8 | |
4 | 3 | subgss 17595 | . . . . . . 7 SubGrp |
5 | 4 | adantl 482 | . . . . . 6 SubGrp SubGrp |
6 | subsubg.h | . . . . . . . 8 ↾s | |
7 | 6 | subgbas 17598 | . . . . . . 7 SubGrp |
8 | 7 | adantr 481 | . . . . . 6 SubGrp SubGrp |
9 | 5, 8 | sseqtr4d 3642 | . . . . 5 SubGrp SubGrp |
10 | eqid 2622 | . . . . . . 7 | |
11 | 10 | subgss 17595 | . . . . . 6 SubGrp |
12 | 11 | adantr 481 | . . . . 5 SubGrp SubGrp |
13 | 9, 12 | sstrd 3613 | . . . 4 SubGrp SubGrp |
14 | 6 | oveq1i 6660 | . . . . . . 7 ↾s ↾s ↾s |
15 | ressabs 15939 | . . . . . . 7 SubGrp ↾s ↾s ↾s | |
16 | 14, 15 | syl5eq 2668 | . . . . . 6 SubGrp ↾s ↾s |
17 | 9, 16 | syldan 487 | . . . . 5 SubGrp SubGrp ↾s ↾s |
18 | eqid 2622 | . . . . . . 7 ↾s ↾s | |
19 | 18 | subggrp 17597 | . . . . . 6 SubGrp ↾s |
20 | 19 | adantl 482 | . . . . 5 SubGrp SubGrp ↾s |
21 | 17, 20 | eqeltrrd 2702 | . . . 4 SubGrp SubGrp ↾s |
22 | 10 | issubg 17594 | . . . 4 SubGrp ↾s |
23 | 2, 13, 21, 22 | syl3anbrc 1246 | . . 3 SubGrp SubGrp SubGrp |
24 | 23, 9 | jca 554 | . 2 SubGrp SubGrp SubGrp |
25 | 6 | subggrp 17597 | . . . 4 SubGrp |
26 | 25 | adantr 481 | . . 3 SubGrp SubGrp |
27 | simprr 796 | . . . 4 SubGrp SubGrp | |
28 | 7 | adantr 481 | . . . 4 SubGrp SubGrp |
29 | 27, 28 | sseqtrd 3641 | . . 3 SubGrp SubGrp |
30 | 16 | adantrl 752 | . . . 4 SubGrp SubGrp ↾s ↾s |
31 | eqid 2622 | . . . . . 6 ↾s ↾s | |
32 | 31 | subggrp 17597 | . . . . 5 SubGrp ↾s |
33 | 32 | ad2antrl 764 | . . . 4 SubGrp SubGrp ↾s |
34 | 30, 33 | eqeltrd 2701 | . . 3 SubGrp SubGrp ↾s |
35 | 3 | issubg 17594 | . . 3 SubGrp ↾s |
36 | 26, 29, 34, 35 | syl3anbrc 1246 | . 2 SubGrp SubGrp SubGrp |
37 | 24, 36 | impbida 877 | 1 SubGrp SubGrp SubGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cgrp 17422 SubGrpcsubg 17588 |
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 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-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
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-ral 2917 df-rex 2918 df-reu 2919 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-subg 17591 |
This theorem is referenced by: nmznsg 17638 subgslw 18031 subgdmdprd 18433 subgdprd 18434 ablfac1c 18470 pgpfaclem1 18480 pgpfaclem2 18481 ablfaclem3 18486 |
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