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Theorem subgdprd 15548
Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
subgdprd.1  |-  H  =  ( Gs  A )
subgdprd.2  |-  ( ph  ->  A  e.  (SubGrp `  G ) )
subgdprd.3  |-  ( ph  ->  G dom DProd  S )
subgdprd.4  |-  ( ph  ->  ran  S  C_  ~P A )
Assertion
Ref Expression
subgdprd  |-  ( ph  ->  ( H DProd  S )  =  ( G DProd  S
) )

Proof of Theorem subgdprd
StepHypRef Expression
1 subgdprd.2 . . . . . 6  |-  ( ph  ->  A  e.  (SubGrp `  G ) )
2 subgdprd.1 . . . . . . 7  |-  H  =  ( Gs  A )
32subggrp 14902 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
41, 3syl 16 . . . . 5  |-  ( ph  ->  H  e.  Grp )
5 eqid 2404 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
65subgacs 14930 . . . . 5  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
7 acsmre 13832 . . . . 5  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
84, 6, 73syl 19 . . . 4  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
9 subgrcl 14904 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
11 eqid 2404 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1211subgacs 14930 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
13 acsmre 13832 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
1410, 12, 133syl 19 . . . . 5  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
15 eqid 2404 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
16 subgdprd.3 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
17 dprdf 15519 . . . . . . . 8  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
18 frn 5556 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
1916, 17, 183syl 19 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
20 mresspw 13772 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2114, 20syl 16 . . . . . . 7  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
2219, 21sstrd 3318 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
23 sspwuni 4136 . . . . . 6  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
2422, 23sylib 189 . . . . 5  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
2514, 15, 24mrcssidd 13805 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
2615mrccl 13791 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  S  C_  ( Base `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
2714, 24, 26syl2anc 643 . . . . 5  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
28 subgdprd.4 . . . . . . 7  |-  ( ph  ->  ran  S  C_  ~P A )
29 sspwuni 4136 . . . . . . 7  |-  ( ran 
S  C_  ~P A  <->  U.
ran  S  C_  A )
3028, 29sylib 189 . . . . . 6  |-  ( ph  ->  U. ran  S  C_  A )
3115mrcsscl 13800 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  S  C_  A  /\  A  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  C_  A
)
3214, 30, 1, 31syl3anc 1184 . . . . 5  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  A )
332subsubg 14918 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  C_  A
) ) )
341, 33syl 16 . . . . 5  |-  ( ph  ->  ( ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
)  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  C_  A
) ) )
3527, 32, 34mpbir2and 889 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
36 eqid 2404 . . . . 5  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
3736mrcsscl 13800 . . . 4  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U.
ran  S  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ran  S
)  /\  ( (mrCls `  (SubGrp `  G )
) `  U. ran  S
)  e.  (SubGrp `  H ) )  -> 
( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
388, 25, 35, 37syl3anc 1184 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
392subgdmdprd 15547 . . . . . . . . . . 11  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
401, 39syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_ 
~P A ) ) )
4116, 28, 40mpbir2and 889 . . . . . . . . 9  |-  ( ph  ->  H dom DProd  S )
42 eqidd 2405 . . . . . . . . 9  |-  ( ph  ->  dom  S  =  dom  S )
4341, 42dprdf2 15520 . . . . . . . 8  |-  ( ph  ->  S : dom  S --> (SubGrp `  H ) )
44 frn 5556 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4543, 44syl 16 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  H ) )
46 mresspw 13772 . . . . . . . 8  |-  ( (SubGrp `  H )  e.  (Moore `  ( Base `  H
) )  ->  (SubGrp `  H )  C_  ~P ( Base `  H )
)
478, 46syl 16 . . . . . . 7  |-  ( ph  ->  (SubGrp `  H )  C_ 
~P ( Base `  H
) )
4845, 47sstrd 3318 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  H )
)
49 sspwuni 4136 . . . . . 6  |-  ( ran 
S  C_  ~P ( Base `  H )  <->  U. ran  S  C_  ( Base `  H
) )
5048, 49sylib 189 . . . . 5  |-  ( ph  ->  U. ran  S  C_  ( Base `  H )
)
518, 36, 50mrcssidd 13805 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
5236mrccl 13791 . . . . . . 7  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U.
ran  S  C_  ( Base `  H ) )  -> 
( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
538, 50, 52syl2anc 643 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
542subsubg 14918 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  C_  A
) ) )
551, 54syl 16 . . . . . 6  |-  ( ph  ->  ( ( (mrCls `  (SubGrp `  H ) ) `
 U. ran  S
)  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  C_  A
) ) )
5653, 55mpbid 202 . . . . 5  |-  ( ph  ->  ( ( (mrCls `  (SubGrp `  H ) ) `
 U. ran  S
)  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  C_  A
) )
5756simpld 446 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
5815mrcsscl 13800 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  S  C_  ( (mrCls `  (SubGrp `  H )
) `  U. ran  S
)  /\  ( (mrCls `  (SubGrp `  H )
) `  U. ran  S
)  e.  (SubGrp `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
5914, 51, 57, 58syl3anc 1184 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
6038, 59eqssd 3325 . 2  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
6136dprdspan 15540 . . 3  |-  ( H dom DProd  S  ->  ( H DProd 
S )  =  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
6241, 61syl 16 . 2  |-  ( ph  ->  ( H DProd  S )  =  ( (mrCls `  (SubGrp `  H ) ) `
 U. ran  S
) )
6315dprdspan 15540 . . 3  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
6416, 63syl 16 . 2  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
6560, 62, 643eqtr4d 2446 1  |-  ( ph  ->  ( H DProd  S )  =  ( G DProd  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   class class class wbr 4172   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425  Moorecmre 13762  mrClscmrc 13763  ACScacs 13765   Grpcgrp 14640  SubGrpcsubg 14893   DProd cdprd 15509
This theorem is referenced by:  ablfaclem3  15600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-gim 15001  df-cntz 15071  df-oppg 15097  df-cmn 15369  df-dprd 15511
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