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Theorem subgdprd 15270
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 14624 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
41, 3syl 15 . . . . 5  |-  ( ph  ->  H  e.  Grp )
5 eqid 2283 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
65subgacs 14652 . . . . 5  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
7 acsmre 13554 . . . . 5  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
84, 6, 73syl 18 . . . 4  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
9 subgrcl 14626 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
101, 9syl 15 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
11 eqid 2283 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1211subgacs 14652 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
13 acsmre 13554 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
1410, 12, 133syl 18 . . . . 5  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
15 subgdprd.3 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
16 dprdf 15241 . . . . . . . 8  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
17 frn 5395 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
1815, 16, 173syl 18 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
19 mresspw 13494 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2014, 19syl 15 . . . . . . 7  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
2118, 20sstrd 3189 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
22 sspwuni 3987 . . . . . 6  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
2321, 22sylib 188 . . . . 5  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
24 eqid 2283 . . . . . 6  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2524mrcssid 13519 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  S  C_  ( Base `  G ) )  ->  U. ran  S  C_  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
2614, 23, 25syl2anc 642 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
2724mrccl 13513 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  S  C_  ( Base `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
2814, 23, 27syl2anc 642 . . . . 5  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
29 subgdprd.4 . . . . . . 7  |-  ( ph  ->  ran  S  C_  ~P A )
30 sspwuni 3987 . . . . . . 7  |-  ( ran 
S  C_  ~P A  <->  U.
ran  S  C_  A )
3129, 30sylib 188 . . . . . 6  |-  ( ph  ->  U. ran  S  C_  A )
3224mrcsscl 13522 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  S  C_  A  /\  A  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  S )  C_  A
)
3314, 31, 1, 32syl3anc 1182 . . . . 5  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  A )
342subsubg 14640 . . . . . 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
) ) )
351, 34syl 15 . . . . 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
) ) )
3628, 33, 35mpbir2and 888 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
37 eqid 2283 . . . . 5  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
3837mrcsscl 13522 . . . 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 ) )
398, 26, 36, 38syl3anc 1182 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
402subgdmdprd 15269 . . . . . . . . . . 11  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
411, 40syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_ 
~P A ) ) )
4215, 29, 41mpbir2and 888 . . . . . . . . 9  |-  ( ph  ->  H dom DProd  S )
43 eqidd 2284 . . . . . . . . 9  |-  ( ph  ->  dom  S  =  dom  S )
4442, 43dprdf2 15242 . . . . . . . 8  |-  ( ph  ->  S : dom  S --> (SubGrp `  H ) )
45 frn 5395 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4644, 45syl 15 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  H ) )
47 mresspw 13494 . . . . . . . 8  |-  ( (SubGrp `  H )  e.  (Moore `  ( Base `  H
) )  ->  (SubGrp `  H )  C_  ~P ( Base `  H )
)
488, 47syl 15 . . . . . . 7  |-  ( ph  ->  (SubGrp `  H )  C_ 
~P ( Base `  H
) )
4946, 48sstrd 3189 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  H )
)
50 sspwuni 3987 . . . . . 6  |-  ( ran 
S  C_  ~P ( Base `  H )  <->  U. ran  S  C_  ( Base `  H
) )
5149, 50sylib 188 . . . . 5  |-  ( ph  ->  U. ran  S  C_  ( Base `  H )
)
5237mrcssid 13519 . . . . 5  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U.
ran  S  C_  ( Base `  H ) )  ->  U. ran  S  C_  (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
538, 51, 52syl2anc 642 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
5437mrccl 13513 . . . . . . 7  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U.
ran  S  C_  ( Base `  H ) )  -> 
( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
558, 51, 54syl2anc 642 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
562subsubg 14640 . . . . . . 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
) ) )
571, 56syl 15 . . . . . 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
) ) )
5855, 57mpbid 201 . . . . 5  |-  ( ph  ->  ( ( (mrCls `  (SubGrp `  H ) ) `
 U. ran  S
)  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  C_  A
) )
5958simpld 445 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
6024mrcsscl 13522 . . . 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 ) )
6114, 53, 59, 60syl3anc 1182 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
6239, 61eqssd 3196 . 2  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
6337dprdspan 15262 . . 3  |-  ( H dom DProd  S  ->  ( H DProd 
S )  =  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
6442, 63syl 15 . 2  |-  ( ph  ->  ( H DProd  S )  =  ( (mrCls `  (SubGrp `  H ) ) `
 U. ran  S
) )
6524dprdspan 15262 . . 3  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
6615, 65syl 15 . 2  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
6762, 64, 663eqtr4d 2325 1  |-  ( ph  ->  ( H DProd  S )  =  ( G DProd  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615   DProd cdprd 15231
This theorem is referenced by:  ablfaclem3  15322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-cmn 15091  df-dprd 15233
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