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Theorem dprdcntz 15243
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 2529 . . 3  |-  ( ph  ->  Y  =/=  X )
4 eldifsn 3749 . . 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 15235 . . . . . . . . . 10  |-  Rel  dom DProd
109brrelex2i 4730 . . . . . . . . 9  |-  ( G dom DProd  S  ->  S  e. 
_V )
11 dmexg 4939 . . . . . . . . 9  |-  ( S  e.  _V  ->  dom  S  e.  _V )
127, 10, 113syl 18 . . . . . . . 8  |-  ( ph  ->  dom  S  e.  _V )
138, 12eqeltrrd 2358 . . . . . . 7  |-  ( ph  ->  I  e.  _V )
14 dprdcntz.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
15 eqid 2283 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
16 eqid 2283 . . . . . . . 8  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
1714, 15, 16dmdprd 15236 . . . . . . 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 2618 . . . 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 3651 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2524difeq2d 3294 . . . . 5  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
26 fveq2 5525 . . . . . 6  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
2726sseq1d 3205 . . . . 5  |-  ( x  =  X  ->  (
( S `  x
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  y ) ) ) )
2825, 27raleqbidv 2748 . . . 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 2880 . . 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 5525 . . . . 5  |-  ( y  =  Y  ->  ( S `  y )  =  ( S `  Y ) )
3231fveq2d 5529 . . . 4  |-  ( y  =  Y  ->  ( Z `  ( S `  y ) )  =  ( Z `  ( S `  Y )
) )
3332sseq2d 3206 . . 3  |-  ( y  =  Y  ->  (
( S `  X
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )
3433rspcv 2880 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   U.cuni 3827   class class class wbr 4023   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255   0gc0g 13400  mrClscmrc 13485   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprdfcntz  15250  dprdfadd  15255  dprdres  15263  dprdss  15264  dprdf1o  15267  dprdcntz2  15273  dprd2da  15277  dmdprdsplit2lem  15280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818  df-dprd 15233
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