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

Theorem subgdprd 15594
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 14948 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
41, 3syl 16 . . . . 5  |-  ( ph  ->  H  e.  Grp )
5 eqid 2437 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
65subgacs 14976 . . . . 5  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
7 acsmre 13878 . . . . 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 14950 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
11 eqid 2437 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1211subgacs 14976 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
13 acsmre 13878 . . . . . 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 2437 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
16 subgdprd.3 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
17 dprdf 15565 . . . . . . . 8  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
18 frn 5598 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
1916, 17, 183syl 19 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
20 mresspw 13818 . . . . . . . 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 3359 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
23 sspwuni 4177 . . . . . 6  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
2422, 23sylib 190 . . . . 5  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
2514, 15, 24mrcssidd 13851 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
2615mrccl 13837 . . . . . 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 644 . . . . 5  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
28 subgdprd.4 . . . . . . 7  |-  ( ph  ->  ran  S  C_  ~P A )
29 sspwuni 4177 . . . . . . 7  |-  ( ran 
S  C_  ~P A  <->  U.
ran  S  C_  A )
3028, 29sylib 190 . . . . . 6  |-  ( ph  ->  U. ran  S  C_  A )
3115mrcsscl 13846 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  A )
332subsubg 14964 . . . . . 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 890 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
36 eqid 2437 . . . . 5  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
3736mrcsscl 13846 . . . 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 1185 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
392subgdmdprd 15593 . . . . . . . . . . 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 890 . . . . . . . . 9  |-  ( ph  ->  H dom DProd  S )
42 eqidd 2438 . . . . . . . . 9  |-  ( ph  ->  dom  S  =  dom  S )
4341, 42dprdf2 15566 . . . . . . . 8  |-  ( ph  ->  S : dom  S --> (SubGrp `  H ) )
44 frn 5598 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4543, 44syl 16 . . . . . . 7  |-  ( ph  ->  ran  S  C_  (SubGrp `  H ) )
46 mresspw 13818 . . . . . . . 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 3359 . . . . . 6  |-  ( ph  ->  ran  S  C_  ~P ( Base `  H )
)
49 sspwuni 4177 . . . . . 6  |-  ( ran 
S  C_  ~P ( Base `  H )  <->  U. ran  S  C_  ( Base `  H
) )
5048, 49sylib 190 . . . . 5  |-  ( ph  ->  U. ran  S  C_  ( Base `  H )
)
518, 36, 50mrcssidd 13851 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
5236mrccl 13837 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  H )
)
542subsubg 14964 . . . . . . 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 203 . . . . 5  |-  ( ph  ->  ( ( (mrCls `  (SubGrp `  H ) ) `
 U. ran  S
)  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ran  S )  C_  A
) )
5756simpld 447 . . . 4  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  e.  (SubGrp `  G )
)
5815mrcsscl 13846 . . . 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 1185 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S )  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ran  S ) )
6038, 59eqssd 3366 . 2  |-  ( ph  ->  ( (mrCls `  (SubGrp `  H ) ) `  U. ran  S )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
6136dprdspan 15586 . . 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 15586 . . 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 2479 1  |-  ( ph  ->  ( H DProd  S )  =  ( G DProd  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3321   ~Pcpw 3800   U.cuni 4016   class class class wbr 4213   dom cdm 4879   ran crn 4880   -->wf 5451   ` cfv 5455  (class class class)co 6082   Basecbs 13470   ↾s cress 13471  Moorecmre 13808  mrClscmrc 13809  ACScacs 13811   Grpcgrp 14686  SubGrpcsubg 14939   DProd cdprd 15555
This theorem is referenced by:  ablfaclem3  15646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-fzo 11137  df-seq 11325  df-hash 11620  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-0g 13728  df-gsum 13729  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-mhm 14739  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-mulg 14816  df-subg 14942  df-ghm 15005  df-gim 15047  df-cntz 15117  df-oppg 15143  df-cmn 15415  df-dprd 15557
  Copyright terms: Public domain W3C validator