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

Theorem dprdcntz 15494
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 2634 . . 3  |-  ( ph  ->  Y  =/=  X )
4 eldifsn 3871 . . 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 15486 . . . . . . . . . 10  |-  Rel  dom DProd
109brrelex2i 4860 . . . . . . . . 9  |-  ( G dom DProd  S  ->  S  e. 
_V )
11 dmexg 5071 . . . . . . . . 9  |-  ( S  e.  _V  ->  dom  S  e.  _V )
127, 10, 113syl 19 . . . . . . . 8  |-  ( ph  ->  dom  S  e.  _V )
138, 12eqeltrrd 2463 . . . . . . 7  |-  ( ph  ->  I  e.  _V )
14 dprdcntz.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
15 eqid 2388 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
16 eqid 2388 . . . . . . . 8  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
1714, 15, 16dmdprd 15487 . . . . . . 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 2725 . . . 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 3769 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2524difeq2d 3409 . . . . 5  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
26 fveq2 5669 . . . . . 6  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
2726sseq1d 3319 . . . . 5  |-  ( x  =  X  ->  (
( S `  x
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  y ) ) ) )
2825, 27raleqbidv 2860 . . . 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 2992 . . 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 5669 . . . . 5  |-  ( y  =  Y  ->  ( S `  y )  =  ( S `  Y ) )
3231fveq2d 5673 . . . 4  |-  ( y  =  Y  ->  ( Z `  ( S `  y ) )  =  ( Z `  ( S `  Y )
) )
3332sseq2d 3320 . . 3  |-  ( y  =  Y  ->  (
( S `  X
)  C_  ( Z `  ( S `  y
) )  <->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )
3433rspcv 2992 . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   _Vcvv 2900    \ cdif 3261    i^i cin 3263    C_ wss 3264   {csn 3758   U.cuni 3958   class class class wbr 4154   dom cdm 4819   "cima 4822   -->wf 5391   ` cfv 5395   0gc0g 13651  mrClscmrc 13736   Grpcgrp 14613  SubGrpcsubg 14866  Cntzccntz 15042   DProd cdprd 15482
This theorem is referenced by:  dprdfcntz  15501  dprdfadd  15506  dprdres  15514  dprdss  15515  dprdf1o  15518  dprdcntz2  15524  dprd2da  15528  dmdprdsplit2lem  15531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-ixp 7001  df-dprd 15484
  Copyright terms: Public domain W3C validator