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Theorem dprdsn 17747
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 2471 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2471 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2471 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 subgrcl 16900 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
54adantl 473 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G  e.  Grp )
6 snex 4641 . . . 4  |-  { A }  e.  _V
76a1i 11 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { A }  e.  _V )
8 f1osng 5867 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } -1-1-onto-> { S } )
9 f1of 5828 . . . . 5  |-  ( {
<. A ,  S >. } : { A } -1-1-onto-> { S }  ->  { <. A ,  S >. } : { A } --> { S } )
108, 9syl 17 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> { S } )
11 simpr 468 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  S  e.  (SubGrp `  G )
)
1211snssd 4108 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { S }  C_  (SubGrp `  G
) )
1310, 12fssd 5750 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> (SubGrp `  G ) )
14 simpr1 1036 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  e.  { A } )
15 elsni 3985 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
1614, 15syl 17 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  A )
17 simpr2 1037 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  e.  { A } )
18 elsni 3985 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
1917, 18syl 17 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  =  A )
2016, 19eqtr4d 2508 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  y )
21 simpr3 1038 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =/=  y )
2220, 21pm2.21ddne 2727 . . 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 ) ) )
235adantr 472 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  G  e.  Grp )
24 eqid 2471 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2524subgacs 16930 . . . . . . . 8  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
26 acsmre 15636 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2723, 25, 263syl 18 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2815adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  x  =  A )
2928sneqd 3971 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { x }  =  { A } )
3029difeq2d 3540 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  ( { A }  \  { A } ) )
31 difid 3747 . . . . . . . . . . . . 13  |-  ( { A }  \  { A } )  =  (/)
3230, 31syl6eq 2521 . . . . . . . . . . . 12  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  (/) )
3332imaeq2d 5174 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  ( { <. A ,  S >. } " (/) ) )
34 ima0 5189 . . . . . . . . . . 11  |-  ( {
<. A ,  S >. }
" (/) )  =  (/)
3533, 34syl6eq 2521 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
3635unieqd 4200 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  U. (/) )
37 uni0 4217 . . . . . . . . 9  |-  U. (/)  =  (/)
3836, 37syl6eq 2521 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
39 0ss 3766 . . . . . . . . 9  |-  (/)  C_  { ( 0g `  G ) }
4039a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (/)  C_  { ( 0g `  G ) } )
4138, 40eqsstrd 3452 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) } )
4220subg 16920 . . . . . . . 8  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
4323, 42syl 17 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
443mrcsscl 15604 . . . . . . 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 ) } )
4527, 41, 43, 44syl3anc 1292 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
462subg0cl 16903 . . . . . . . . 9  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
4746ad2antlr 741 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  S )
4815fveq2d 5883 . . . . . . . . 9  |-  ( x  e.  { A }  ->  ( { <. A ,  S >. } `  x
)  =  ( {
<. A ,  S >. } `
 A ) )
49 fvsng 6114 . . . . . . . . 9  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( { <. A ,  S >. } `  A )  =  S )
5048, 49sylan9eqr 2527 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } `  x )  =  S )
5147, 50eleqtrrd 2552 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  ( { <. A ,  S >. } `  x
) )
5251snssd 4108 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  C_  ( { <. A ,  S >. } `
 x ) )
5345, 52sstrd 3428 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  ( { <. A ,  S >. } `
 x ) )
54 dfss1 3628 . . . . 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 }
) ) ) )
5553, 54sylib 201 . . . 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 }
) ) ) )
5655, 45eqsstrd 3452 . . 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 ) } )
571, 2, 3, 5, 7, 13, 22, 56dmdprdd 17709 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G dom DProd  { <. A ,  S >. } )
583dprdspan 17738 . . . 4  |-  ( G dom DProd  { <. A ,  S >. }  ->  ( G DProd  {
<. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  { <. A ,  S >. } ) )
5957, 58syl 17 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } ) )
60 rnsnopg 5322 . . . . . . . 8  |-  ( A  e.  V  ->  ran  {
<. A ,  S >. }  =  { S }
)
6160adantr 472 . . . . . . 7  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ran  {
<. A ,  S >. }  =  { S }
)
6261unieqd 4200 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  U. { S } )
63 unisng 4206 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  U. { S }  =  S )
6463adantl 473 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. { S }  =  S
)
6562, 64eqtrd 2505 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  S )
6665fveq2d 5883 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  S
) )
675, 25, 263syl 18 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
683mrcid 15597 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
6967, 68sylancom 680 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7066, 69eqtrd 2505 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  S )
7159, 70eqtrd 2505 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  S )
7257, 71jca 541 1  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395   dom cdm 4839   ran crn 4840   "cima 4842   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   Basecbs 15199   0gc0g 15416  Moorecmre 15566  mrClscmrc 15567  ACScacs 15569   Grpcgrp 16747  SubGrpcsubg 16889  Cntzccntz 17047   DProd cdprd 17703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-gim 17001  df-cntz 17049  df-oppg 17075  df-cmn 17510  df-dprd 17705
This theorem is referenced by:  dprd2da  17753  dmdprdpr  17760  dprdpr  17761  dpjlem  17762  pgpfaclem1  17792
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