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

Theorem dprdsn 16654
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 2454 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2454 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2454 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 subgrcl 15804 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
54adantl 466 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G  e.  Grp )
6 snex 4640 . . . 4  |-  { A }  e.  _V
76a1i 11 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { A }  e.  _V )
8 f1osng 5786 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } -1-1-onto-> { S } )
9 f1of 5748 . . . . 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 461 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  S  e.  (SubGrp `  G )
)
1211snssd 4125 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { S }  C_  (SubGrp `  G
) )
13 fss 5674 . . . 4  |-  ( ( { <. A ,  S >. } : { A }
--> { S }  /\  { S }  C_  (SubGrp `  G ) )  ->  { <. A ,  S >. } : { A }
--> (SubGrp `  G )
)
1410, 12, 13syl2anc 661 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> (SubGrp `  G ) )
15 simpr1 994 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  e.  { A } )
16 elsni 4009 . . . . . 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 995 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  e.  { A } )
19 elsni 4009 . . . . . 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 2498 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  y )
22 simpr3 996 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =/=  y )
2321, 22pm2.21ddne 2765 . . 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 465 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  G  e.  Grp )
25 eqid 2454 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgacs 15834 . . . . . . . 8  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
27 acsmre 14708 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2824, 26, 273syl 20 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2916adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  x  =  A )
3029sneqd 3996 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { x }  =  { A } )
3130difeq2d 3581 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  ( { A }  \  { A } ) )
32 difid 3854 . . . . . . . . . . . . 13  |-  ( { A }  \  { A } )  =  (/)
3331, 32syl6eq 2511 . . . . . . . . . . . 12  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  (/) )
3433imaeq2d 5276 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  ( { <. A ,  S >. } " (/) ) )
35 ima0 5291 . . . . . . . . . . 11  |-  ( {
<. A ,  S >. }
" (/) )  =  (/)
3634, 35syl6eq 2511 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
3736unieqd 4208 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  U. (/) )
38 uni0 4225 . . . . . . . . 9  |-  U. (/)  =  (/)
3937, 38syl6eq 2511 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
40 0ss 3773 . . . . . . . . 9  |-  (/)  C_  { ( 0g `  G ) }
4140a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (/)  C_  { ( 0g `  G ) } )
4239, 41eqsstrd 3497 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) } )
4320subg 15824 . . . . . . . 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 14676 . . . . . . 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 1219 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
472subg0cl 15807 . . . . . . . . 9  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
4847ad2antlr 726 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  S )
4916fveq2d 5802 . . . . . . . . 9  |-  ( x  e.  { A }  ->  ( { <. A ,  S >. } `  x
)  =  ( {
<. A ,  S >. } `
 A ) )
50 fvsng 6020 . . . . . . . . 9  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( { <. A ,  S >. } `  A )  =  S )
5149, 50sylan9eqr 2517 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } `  x )  =  S )
5248, 51eleqtrrd 2545 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  ( { <. A ,  S >. } `  x
) )
5352snssd 4125 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  C_  ( { <. A ,  S >. } `
 x ) )
5446, 53sstrd 3473 . . . . 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 3662 . . . . 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 196 . . . 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 3497 . . 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 16602 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G dom DProd  { <. A ,  S >. } )
593dprdspan 16645 . . . 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 5425 . . . . . . . 8  |-  ( A  e.  V  ->  ran  {
<. A ,  S >. }  =  { S }
)
6261adantr 465 . . . . . . 7  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ran  {
<. A ,  S >. }  =  { S }
)
6362unieqd 4208 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  U. { S } )
64 unisng 4214 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  U. { S }  =  S )
6564adantl 466 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. { S }  =  S
)
6663, 65eqtrd 2495 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  S )
6766fveq2d 5802 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  S
) )
685, 26, 273syl 20 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
693mrcid 14669 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7068, 11, 69syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7167, 70eqtrd 2495 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  S )
7260, 71eqtrd 2495 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  S )
7358, 72jca 532 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   _Vcvv 3076    \ cdif 3432    i^i cin 3434    C_ wss 3435   (/)c0 3744   {csn 3984   <.cop 3990   U.cuni 4198   class class class wbr 4399   dom cdm 4947   ran crn 4948   "cima 4950   -->wf 5521   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199   Basecbs 14291   0gc0g 14496  Moorecmre 14638  mrClscmrc 14639  ACScacs 14641   Grpcgrp 15528  SubGrpcsubg 15793  Cntzccntz 15951   DProd cdprd 16596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-tpos 6854  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-seq 11923  df-hash 12220  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-0g 14498  df-gsum 14499  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-mhm 15582  df-submnd 15583  df-grp 15663  df-minusg 15664  df-sbg 15665  df-mulg 15666  df-subg 15796  df-ghm 15863  df-gim 15905  df-cntz 15953  df-oppg 15979  df-cmn 16399  df-dprd 16598
This theorem is referenced by:  dprd2da  16662  dmdprdpr  16669  dprdpr  16670  dpjlem  16671  pgpfaclem1  16703
  Copyright terms: Public domain W3C validator