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Theorem dprdcntz 15259
Description: The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdcntz.1  |-  ( ph  ->  G dom DProd  S )
dprdcntz.2  |-  ( ph  ->  dom  S  =  I )
dprdcntz.3  |-  ( ph  ->  X  e.  I )
dprdcntz.4  |-  ( ph  ->  Y  e.  I )
dprdcntz.5  |-  ( ph  ->  X  =/=  Y )
dprdcntz.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
dprdcntz  |-  ( ph  ->  ( S `  X
)  C_  ( Z `  ( S `  Y
) ) )

Proof of Theorem dprdcntz
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdcntz.4 . . 3  |-  ( ph  ->  Y  e.  I )
2 dprdcntz.5 . . . 4  |-  ( ph  ->  X  =/=  Y )
32necomd 2542 . . 3  |-  ( ph  ->  Y  =/=  X )
4 eldifsn 3762 . . 3  |-  ( Y  e.  ( I  \  { X } )  <->  ( Y  e.  I  /\  Y  =/= 
X ) )
51, 3, 4sylanbrc 645 . 2  |-  ( ph  ->  Y  e.  ( I 
\  { X }
) )
6 dprdcntz.3 . . 3  |-  ( ph  ->  X  e.  I )
7 dprdcntz.1 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
8 dprdcntz.2 . . . . . . . 8  |-  ( ph  ->  dom  S  =  I )
9 reldmdprd 15251 . . . . . . . . . 10  |-  Rel  dom DProd
109brrelex2i 4746 . . . . . . . . 9  |-  ( G dom DProd  S  ->  S  e. 
_V )
11 dmexg 4955 . . . . . . . . 9  |-  ( S  e.  _V  ->  dom  S  e.  _V )
127, 10, 113syl 18 . . . . . . . 8  |-  ( ph  ->  dom  S  e.  _V )
138, 12eqeltrrd 2371 . . . . . . 7  |-  ( ph  ->  I  e.  _V )
14 dprdcntz.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
15 eqid 2296 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
16 eqid 2296 . . . . . . . 8  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
1714, 15, 16dmdprd 15252 . . . . . . 7  |-  ( ( I  e.  _V  /\  dom  S  =  I )  ->  ( G dom DProd  S  <-> 
( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
1813, 8, 17syl2anc 642 . . . . . 6  |-  ( ph  ->  ( G dom DProd  S  <->  ( G  e.  Grp  /\  S :
I --> (SubGrp `  G )  /\  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
197, 18mpbid 201 . . . . 5  |-  ( ph  ->  ( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) )
2019simp3d 969 . . . 4  |-  ( ph  ->  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )
21 simpl 443 . . . . 5  |-  ( ( A. y  e.  ( I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )  ->  A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) ) )
2221ralimi 2631 . . . 4  |-  ( A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )  ->  A. x  e.  I  A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) ) )
2320, 22syl 15 . . 3  |-  ( ph  ->  A. x  e.  I  A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) ) )
24 sneq 3664 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2524difeq2d 3307 . . . . 5  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
26 fveq2 5541 . . . . . 6  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
2726sseq1d 3218 . . . . 5  |-  ( x  =  X  ->  (
( S `  x
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  y ) ) ) )
2825, 27raleqbidv 2761 . . . 4  |-  ( x  =  X  ->  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  <->  A. y  e.  (
I  \  { X } ) ( S `
 X )  C_  ( Z `  ( S `
 y ) ) ) )
2928rspcv 2893 . . 3  |-  ( X  e.  I  ->  ( A. x  e.  I  A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  ->  A. y  e.  ( I  \  { X } ) ( S `
 X )  C_  ( Z `  ( S `
 y ) ) ) )
306, 23, 29sylc 56 . 2  |-  ( ph  ->  A. y  e.  ( I  \  { X } ) ( S `
 X )  C_  ( Z `  ( S `
 y ) ) )
31 fveq2 5541 . . . . 5  |-  ( y  =  Y  ->  ( S `  y )  =  ( S `  Y ) )
3231fveq2d 5545 . . . 4  |-  ( y  =  Y  ->  ( Z `  ( S `  y ) )  =  ( Z `  ( S `  Y )
) )
3332sseq2d 3219 . . 3  |-  ( y  =  Y  ->  (
( S `  X
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )
3433rspcv 2893 . 2  |-  ( Y  e.  ( I  \  { X } )  -> 
( A. y  e.  ( I  \  { X } ) ( S `
 X )  C_  ( Z `  ( S `
 y ) )  ->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )
355, 30, 34sylc 56 1  |-  ( ph  ->  ( S `  X
)  C_  ( Z `  ( S `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653   U.cuni 3843   class class class wbr 4039   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271   0gc0g 13416  mrClscmrc 13501   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdfcntz  15266  dprdfadd  15271  dprdres  15279  dprdss  15280  dprdf1o  15283  dprdcntz2  15289  dprd2da  15293  dmdprdsplit2lem  15296
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-reu 2563  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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ixp 6834  df-dprd 15249
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