MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgdprd Unicode version

Theorem subgdprd 15286
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 14640 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
41, 3syl 15 . . . . 5  |-  ( ph  ->  H  e.  Grp )
5 eqid 2296 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
65subgacs 14668 . . . . 5  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
7 acsmre 13570 . . . . 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 14642 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
101, 9syl 15 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
11 eqid 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1211subgacs 14668 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
13 acsmre 13570 . . . . . 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 15257 . . . . . . . 8  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
17 frn 5411 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
1815, 16, 173syl 18 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
19 mresspw 13510 . . . . . . . 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 3202 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
22 sspwuni 4003 . . . . . 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 2296 . . . . . 6  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2524mrcssid 13535 . . . . 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 13529 . . . . . 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 4003 . . . . . . 7  |-  ( ran 
S  C_  ~P A  <->  U.
ran  S  C_  A )
3129, 30sylib 188 . . . . . 6  |-  ( ph  ->  U. ran  S  C_  A )
3224mrcsscl 13538 . . . . . 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 14656 . . . . . 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 2296 . . . . 5  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
3837mrcsscl 13538 . . . 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 15285 . . . . . . . . . . 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 2297 . . . . . . . . 9  |-  ( ph  ->  dom  S  =  dom  S )
4442, 43dprdf2 15258 . . . . . . . 8  |-  ( ph  ->  S : dom  S --> (SubGrp `  H ) )
45 frn 5411 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4644, 45syl 15 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  H ) )
47 mresspw 13510 . . . . . . . 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 3202 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  H )
)
50 sspwuni 4003 . . . . . 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 13535 . . . . 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 13529 . . . . . . 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 14656 . . . . . . 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 13538 . . . 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 3209 . 2  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
6337dprdspan 15278 . . 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 15278 . . 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 2338 1  |-  ( ph  ->  ( H DProd  S )  =  ( G DProd  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631   DProd cdprd 15247
This theorem is referenced by:  ablfaclem3  15338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-gim 14739  df-cntz 14809  df-oppg 14835  df-cmn 15107  df-dprd 15249
  Copyright terms: Public domain W3C validator