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Theorem dprdcntz 15556
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 2681 . . 3  |-  ( ph  ->  Y  =/=  X )
4 eldifsn 3919 . . 3  |-  ( Y  e.  ( I  \  { X } )  <->  ( Y  e.  I  /\  Y  =/= 
X ) )
51, 3, 4sylanbrc 646 . 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 15548 . . . . . . . . . 10  |-  Rel  dom DProd
109brrelex2i 4911 . . . . . . . . 9  |-  ( G dom DProd  S  ->  S  e. 
_V )
11 dmexg 5122 . . . . . . . . 9  |-  ( S  e.  _V  ->  dom  S  e.  _V )
127, 10, 113syl 19 . . . . . . . 8  |-  ( ph  ->  dom  S  e.  _V )
138, 12eqeltrrd 2510 . . . . . . 7  |-  ( ph  ->  I  e.  _V )
14 dprdcntz.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
15 eqid 2435 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
16 eqid 2435 . . . . . . . 8  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
1714, 15, 16dmdprd 15549 . . . . . . 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 643 . . . . . 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 202 . . . . 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 971 . . . 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 444 . . . . 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 2773 . . . 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 16 . . 3  |-  ( ph  ->  A. x  e.  I  A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) ) )
24 sneq 3817 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2524difeq2d 3457 . . . . 5  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
26 fveq2 5720 . . . . . 6  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
2726sseq1d 3367 . . . . 5  |-  ( x  =  X  ->  (
( S `  x
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  y ) ) ) )
2825, 27raleqbidv 2908 . . . 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 3040 . . 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 58 . 2  |-  ( ph  ->  A. y  e.  ( I  \  { X } ) ( S `
 X )  C_  ( Z `  ( S `
 y ) ) )
31 fveq2 5720 . . . . 5  |-  ( y  =  Y  ->  ( S `  y )  =  ( S `  Y ) )
3231fveq2d 5724 . . . 4  |-  ( y  =  Y  ->  ( Z `  ( S `  y ) )  =  ( Z `  ( S `  Y )
) )
3332sseq2d 3368 . . 3  |-  ( y  =  Y  ->  (
( S `  X
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )
3433rspcv 3040 . 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 58 1  |-  ( ph  ->  ( S `  X
)  C_  ( Z `  ( S `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   U.cuni 4007   class class class wbr 4204   dom cdm 4870   "cima 4873   -->wf 5442   ` cfv 5446   0gc0g 13713  mrClscmrc 13798   Grpcgrp 14675  SubGrpcsubg 14928  Cntzccntz 15104   DProd cdprd 15544
This theorem is referenced by:  dprdfcntz  15563  dprdfadd  15568  dprdres  15576  dprdss  15577  dprdf1o  15580  dprdcntz2  15586  dprd2da  15590  dmdprdsplit2lem  15593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-ixp 7056  df-dprd 15546
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