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Theorem dprdsn 15549
Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdsn  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )

Proof of Theorem dprdsn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2404 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2404 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 subgrcl 14904 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
54adantl 453 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G  e.  Grp )
6 snex 4365 . . . 4  |-  { A }  e.  _V
76a1i 11 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { A }  e.  _V )
8 f1osng 5675 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } -1-1-onto-> { S } )
9 f1of 5633 . . . . 5  |-  ( {
<. A ,  S >. } : { A } -1-1-onto-> { S }  ->  { <. A ,  S >. } : { A } --> { S } )
108, 9syl 16 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> { S } )
11 simpr 448 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  S  e.  (SubGrp `  G )
)
1211snssd 3903 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { S }  C_  (SubGrp `  G
) )
13 fss 5558 . . . 4  |-  ( ( { <. A ,  S >. } : { A }
--> { S }  /\  { S }  C_  (SubGrp `  G ) )  ->  { <. A ,  S >. } : { A }
--> (SubGrp `  G )
)
1410, 12, 13syl2anc 643 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> (SubGrp `  G ) )
15 simpr1 963 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  e.  { A } )
16 elsni 3798 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
1715, 16syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  A )
18 simpr2 964 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  e.  { A } )
19 elsni 3798 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
2018, 19syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  =  A )
2117, 20eqtr4d 2439 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  y )
22 simpr3 965 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =/=  y )
2321, 22pm2.21ddne 2641 . . 3  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
( { <. A ,  S >. } `  x
)  C_  ( (Cntz `  G ) `  ( { <. A ,  S >. } `  y ) ) )
245adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  G  e.  Grp )
25 eqid 2404 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgacs 14930 . . . . . . . 8  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
27 acsmre 13832 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2824, 26, 273syl 19 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2916adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  x  =  A )
3029sneqd 3787 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { x }  =  { A } )
3130difeq2d 3425 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  ( { A }  \  { A } ) )
32 difid 3656 . . . . . . . . . . . . 13  |-  ( { A }  \  { A } )  =  (/)
3331, 32syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  (/) )
3433imaeq2d 5162 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  ( { <. A ,  S >. } " (/) ) )
35 ima0 5180 . . . . . . . . . . 11  |-  ( {
<. A ,  S >. }
" (/) )  =  (/)
3634, 35syl6eq 2452 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
3736unieqd 3986 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  U. (/) )
38 uni0 4002 . . . . . . . . 9  |-  U. (/)  =  (/)
3937, 38syl6eq 2452 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
40 0ss 3616 . . . . . . . . 9  |-  (/)  C_  { ( 0g `  G ) }
4140a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (/)  C_  { ( 0g `  G ) } )
4239, 41eqsstrd 3342 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) } )
4320subg 14920 . . . . . . . 8  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
4424, 43syl 16 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
453mrcsscl 13800 . . . . . . 7  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) }  /\  { ( 0g
`  G ) }  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
4628, 42, 44, 45syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
472subg0cl 14907 . . . . . . . . 9  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
4847ad2antlr 708 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  S )
4916fveq2d 5691 . . . . . . . . 9  |-  ( x  e.  { A }  ->  ( { <. A ,  S >. } `  x
)  =  ( {
<. A ,  S >. } `
 A ) )
50 fvsng 5886 . . . . . . . . 9  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( { <. A ,  S >. } `  A )  =  S )
5149, 50sylan9eqr 2458 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } `  x )  =  S )
5248, 51eleqtrrd 2481 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  ( { <. A ,  S >. } `  x
) )
5352snssd 3903 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  C_  ( { <. A ,  S >. } `
 x ) )
5446, 53sstrd 3318 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  ( { <. A ,  S >. } `
 x ) )
55 dfss1 3505 . . . . 5  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  ( { <. A ,  S >. } `
 x )  <->  ( ( { <. A ,  S >. } `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )
5654, 55sylib 189 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
( { <. A ,  S >. } `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )
5756, 46eqsstrd 3342 . . 3  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
( { <. A ,  S >. } `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) ) 
C_  { ( 0g
`  G ) } )
581, 2, 3, 5, 7, 14, 23, 57dmdprdd 15515 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G dom DProd  { <. A ,  S >. } )
593dprdspan 15540 . . . 4  |-  ( G dom DProd  { <. A ,  S >. }  ->  ( G DProd  {
<. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  { <. A ,  S >. } ) )
6058, 59syl 16 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } ) )
61 rnsnopg 5308 . . . . . . . 8  |-  ( A  e.  V  ->  ran  {
<. A ,  S >. }  =  { S }
)
6261adantr 452 . . . . . . 7  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ran  {
<. A ,  S >. }  =  { S }
)
6362unieqd 3986 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  U. { S } )
64 unisng 3992 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  U. { S }  =  S )
6564adantl 453 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. { S }  =  S
)
6663, 65eqtrd 2436 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  S )
6766fveq2d 5691 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  S
) )
685, 26, 273syl 19 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
693mrcid 13793 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7068, 11, 69syl2anc 643 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7167, 70eqtrd 2436 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  S )
7260, 71eqtrd 2436 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  S )
7358, 72jca 519 1  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   U.cuni 3975   class class class wbr 4172   dom cdm 4837   ran crn 4838   "cima 4840   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Basecbs 13424   0gc0g 13678  Moorecmre 13762  mrClscmrc 13763  ACScacs 13765   Grpcgrp 14640  SubGrpcsubg 14893  Cntzccntz 15069   DProd cdprd 15509
This theorem is referenced by:  dprd2da  15555  dmdprdpr  15562  dprdpr  15563  dpjlem  15564  pgpfaclem1  15594
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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