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Theorem mthmpps 30220
Description: Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many dv conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmpps.r  |-  R  =  (mStRed `  T )
mthmpps.j  |-  J  =  (mPPSt `  T )
mthmpps.u  |-  U  =  (mThm `  T )
mthmpps.d  |-  D  =  (mDV `  T )
mthmpps.v  |-  V  =  (mVars `  T )
mthmpps.z  |-  Z  = 
U. ( V "
( H  u.  { A } ) )
mthmpps.m  |-  M  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) )
Assertion
Ref Expression
mthmpps  |-  ( T  e. mFS  ->  ( <. C ,  H ,  A >.  e.  U  <->  ( <. M ,  H ,  A >.  e.  J  /\  ( R `
 <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) ) ) )

Proof of Theorem mthmpps
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mthmpps.m . . . . . . . 8  |-  M  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) )
2 mthmpps.u . . . . . . . . . . . . . 14  |-  U  =  (mThm `  T )
3 eqid 2451 . . . . . . . . . . . . . 14  |-  (mPreSt `  T )  =  (mPreSt `  T )
42, 3mthmsta 30216 . . . . . . . . . . . . 13  |-  U  C_  (mPreSt `  T )
5 simpr 463 . . . . . . . . . . . . 13  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  U )
64, 5sseldi 3430 . . . . . . . . . . . 12  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  (mPreSt `  T ) )
7 mthmpps.d . . . . . . . . . . . . 13  |-  D  =  (mDV `  T )
8 eqid 2451 . . . . . . . . . . . . 13  |-  (mEx `  T )  =  (mEx
`  T )
97, 8, 3elmpst 30174 . . . . . . . . . . . 12  |-  ( <. C ,  H ,  A >.  e.  (mPreSt `  T )  <->  ( ( C  C_  D  /\  `' C  =  C )  /\  ( H  C_  (mEx `  T )  /\  H  e.  Fin )  /\  A  e.  (mEx `  T )
) )
106, 9sylib 200 . . . . . . . . . . 11  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( ( C  C_  D  /\  `' C  =  C )  /\  ( H  C_  (mEx `  T
)  /\  H  e.  Fin )  /\  A  e.  (mEx `  T )
) )
1110simp1d 1020 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  C_  D  /\  `' C  =  C
) )
1211simpld 461 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  C  C_  D )
13 difssd 3561 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( D  \  ( Z  X.  Z ) ) 
C_  D )
1412, 13unssd 3610 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  u.  ( D  \  ( Z  X.  Z ) ) ) 
C_  D )
151, 14syl5eqss 3476 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  M  C_  D )
1611simprd 465 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' C  =  C
)
17 cnvdif 5242 . . . . . . . . . . 11  |-  `' ( D  \  ( Z  X.  Z ) )  =  ( `' D  \  `' ( Z  X.  Z ) )
18 cnvdif 5242 . . . . . . . . . . . . . 14  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  ( `' ( (mVR `  T )  X.  (mVR `  T ) )  \  `'  _I  )
19 cnvxp 5254 . . . . . . . . . . . . . . 15  |-  `' ( (mVR `  T )  X.  (mVR `  T )
)  =  ( (mVR
`  T )  X.  (mVR `  T )
)
20 cnvi 5240 . . . . . . . . . . . . . . 15  |-  `'  _I  =  _I
2119, 20difeq12i 3549 . . . . . . . . . . . . . 14  |-  ( `' ( (mVR `  T
)  X.  (mVR `  T ) )  \  `'  _I  )  =  ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
2218, 21eqtri 2473 . . . . . . . . . . . . 13  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  (
( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
23 eqid 2451 . . . . . . . . . . . . . . 15  |-  (mVR `  T )  =  (mVR
`  T )
2423, 7mdvval 30142 . . . . . . . . . . . . . 14  |-  D  =  ( ( (mVR `  T )  X.  (mVR `  T ) )  \  _I  )
2524cnveqi 5009 . . . . . . . . . . . . 13  |-  `' D  =  `' ( ( (mVR
`  T )  X.  (mVR `  T )
)  \  _I  )
2622, 25, 243eqtr4i 2483 . . . . . . . . . . . 12  |-  `' D  =  D
27 cnvxp 5254 . . . . . . . . . . . 12  |-  `' ( Z  X.  Z )  =  ( Z  X.  Z )
2826, 27difeq12i 3549 . . . . . . . . . . 11  |-  ( `' D  \  `' ( Z  X.  Z ) )  =  ( D 
\  ( Z  X.  Z ) )
2917, 28eqtri 2473 . . . . . . . . . 10  |-  `' ( D  \  ( Z  X.  Z ) )  =  ( D  \ 
( Z  X.  Z
) )
3029a1i 11 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' ( D  \ 
( Z  X.  Z
) )  =  ( D  \  ( Z  X.  Z ) ) )
3116, 30uneq12d 3589 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
321cnveqi 5009 . . . . . . . . 9  |-  `' M  =  `' ( C  u.  ( D  \  ( Z  X.  Z ) ) )
33 cnvun 5241 . . . . . . . . 9  |-  `' ( C  u.  ( D 
\  ( Z  X.  Z ) ) )  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3432, 33eqtri 2473 . . . . . . . 8  |-  `' M  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3531, 34, 13eqtr4g 2510 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' M  =  M
)
3615, 35jca 535 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( M  C_  D  /\  `' M  =  M
) )
3710simp2d 1021 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( H  C_  (mEx `  T )  /\  H  e.  Fin ) )
3810simp3d 1022 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  (mEx `  T
) )
397, 8, 3elmpst 30174 . . . . . 6  |-  ( <. M ,  H ,  A >.  e.  (mPreSt `  T )  <->  ( ( M  C_  D  /\  `' M  =  M )  /\  ( H  C_  (mEx `  T )  /\  H  e.  Fin )  /\  A  e.  (mEx `  T )
) )
4036, 37, 38, 39syl3anbrc 1192 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. M ,  H ,  A >.  e.  (mPreSt `  T ) )
41 mthmpps.r . . . . . . . 8  |-  R  =  (mStRed `  T )
42 mthmpps.j . . . . . . . 8  |-  J  =  (mPPSt `  T )
4341, 42, 2elmthm 30214 . . . . . . 7  |-  ( <. C ,  H ,  A >.  e.  U  <->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
445, 43sylib 200 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
45 eqid 2451 . . . . . . . 8  |-  (mCls `  T )  =  (mCls `  T )
46 simpll 760 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  T  e. mFS )
4715adantr 467 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  M  C_  D )
4837simpld 461 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  H  C_  (mEx `  T
) )
4948adantr 467 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  H  C_  (mEx `  T
) )
503, 42mppspst 30212 . . . . . . . . . . . . . . . . . . 19  |-  J  C_  (mPreSt `  T )
51 simprl 764 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  J )
5250, 51sseldi 3430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  (mPreSt `  T
) )
533mpst123 30178 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  (mPreSt `  T
)  ->  x  =  <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )
5452, 53syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  =  <. ( 1st `  ( 1st `  x
) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )
5554fveq2d 5869 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( R `  x
)  =  ( R `
 <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. ) )
56 simprr 766 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( R `  x
)  =  ( R `
 <. C ,  H ,  A >. ) )
5755, 56eqtr3d 2487 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( R `  <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )  =  ( R `  <. C ,  H ,  A >. ) )
5854, 52eqeltrrd 2530 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >.  e.  (mPreSt `  T ) )
59 mthmpps.v . . . . . . . . . . . . . . . . 17  |-  V  =  (mVars `  T )
60 eqid 2451 . . . . . . . . . . . . . . . . 17  |-  U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  =  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )
6159, 3, 41, 60msrval 30176 . . . . . . . . . . . . . . . 16  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >.  e.  (mPreSt `  T )  ->  ( R `  <. ( 1st `  ( 1st `  x
) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )  =  <. ( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )
6258, 61syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( R `  <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )  =  <. ( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )
63 mthmpps.z . . . . . . . . . . . . . . . . . 18  |-  Z  = 
U. ( V "
( H  u.  { A } ) )
6459, 3, 41, 63msrval 30176 . . . . . . . . . . . . . . . . 17  |-  ( <. C ,  H ,  A >.  e.  (mPreSt `  T )  ->  ( R `  <. C ,  H ,  A >. )  =  <. ( C  i^i  ( Z  X.  Z
) ) ,  H ,  A >. )
656, 64syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. C ,  H ,  A >. )  =  <. ( C  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
6665adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( R `  <. C ,  H ,  A >. )  =  <. ( C  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
6757, 62, 663eqtr3d 2493 . . . . . . . . . . . . . 14  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  <. ( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >.  =  <. ( C  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
68 fvex 5875 . . . . . . . . . . . . . . . 16  |-  ( 1st `  ( 1st `  x
) )  e.  _V
6968inex1 4544 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) )  e.  _V
70 fvex 5875 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( 1st `  x
) )  e.  _V
71 fvex 5875 . . . . . . . . . . . . . . 15  |-  ( 2nd `  x )  e.  _V
7269, 70, 71otth 4684 . . . . . . . . . . . . . 14  |-  ( <.
( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >.  =  <. ( C  i^i  ( Z  X.  Z ) ) ,  H ,  A >.  <->  (
( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) )  =  ( C  i^i  ( Z  X.  Z ) )  /\  ( 2nd `  ( 1st `  x ) )  =  H  /\  ( 2nd `  x )  =  A ) )
7367, 72sylib 200 . . . . . . . . . . . . 13  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( ( 1st `  ( 1st `  x
) )  i^i  ( U. ( V " (
( 2nd `  ( 1st `  x ) )  u.  { ( 2nd `  x ) } ) )  X.  U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) )  =  ( C  i^i  ( Z  X.  Z ) )  /\  ( 2nd `  ( 1st `  x ) )  =  H  /\  ( 2nd `  x )  =  A ) )
7473simp1d 1020 . . . . . . . . . . . 12  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) )  =  ( C  i^i  ( Z  X.  Z ) ) )
7573simp2d 1021 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  ( 1st `  x ) )  =  H )
7673simp3d 1022 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  x
)  =  A )
7776sneqd 3980 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  { ( 2nd `  x
) }  =  { A } )
7875, 77uneq12d 3589 . . . . . . . . . . . . . . . . 17  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( 2nd `  ( 1st `  x ) )  u.  { ( 2nd `  x ) } )  =  ( H  u.  { A } ) )
7978imaeq2d 5168 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( V " (
( 2nd `  ( 1st `  x ) )  u.  { ( 2nd `  x ) } ) )  =  ( V
" ( H  u.  { A } ) ) )
8079unieqd 4208 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  U. ( V " (
( 2nd `  ( 1st `  x ) )  u.  { ( 2nd `  x ) } ) )  =  U. ( V " ( H  u.  { A } ) ) )
8180, 63syl6eqr 2503 . . . . . . . . . . . . . 14  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  U. ( V " (
( 2nd `  ( 1st `  x ) )  u.  { ( 2nd `  x ) } ) )  =  Z )
8281sqxpeqd 4860 . . . . . . . . . . . . 13  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) )  =  ( Z  X.  Z ) )
8382ineq2d 3634 . . . . . . . . . . . 12  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( 1st `  ( 1st `  x ) )  i^i  ( U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  X.  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) ) ) )  =  ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) ) )
8474, 83eqtr3d 2487 . . . . . . . . . . 11  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( C  i^i  ( Z  X.  Z ) )  =  ( ( 1st `  ( 1st `  x
) )  i^i  ( Z  X.  Z ) ) )
85 inss1 3652 . . . . . . . . . . 11  |-  ( C  i^i  ( Z  X.  Z ) )  C_  C
8684, 85syl6eqssr 3483 . . . . . . . . . 10  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  C_  C )
87 eqidd 2452 . . . . . . . . . . . . . . 15  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 1st `  ( 1st `  x ) )  =  ( 1st `  ( 1st `  x ) ) )
8887, 75, 76oteq123d 4181 . . . . . . . . . . . . . 14  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >.  =  <. ( 1st `  ( 1st `  x
) ) ,  H ,  A >. )
8954, 88eqtrd 2485 . . . . . . . . . . . . 13  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  =  <. ( 1st `  ( 1st `  x
) ) ,  H ,  A >. )
9089, 52eqeltrrd 2530 . . . . . . . . . . . 12  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  <. ( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
) )
917, 8, 3elmpst 30174 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  <->  ( ( ( 1st `  ( 1st `  x ) )  C_  D  /\  `' ( 1st `  ( 1st `  x
) )  =  ( 1st `  ( 1st `  x ) ) )  /\  ( H  C_  (mEx `  T )  /\  H  e.  Fin )  /\  A  e.  (mEx `  T ) ) )
9291simp1bi 1023 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( ( 1st `  ( 1st `  x
) )  C_  D  /\  `' ( 1st `  ( 1st `  x ) )  =  ( 1st `  ( 1st `  x ) ) ) )
9392simpld 461 . . . . . . . . . . . 12  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( 1st `  ( 1st `  x
) )  C_  D
)
9490, 93syl 17 . . . . . . . . . . 11  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 1st `  ( 1st `  x ) ) 
C_  D )
9594ssdifd 3569 . . . . . . . . . 10  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) )  C_  ( D  \  ( Z  X.  Z ) ) )
96 unss12 3606 . . . . . . . . . 10  |-  ( ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  C_  C  /\  ( ( 1st `  ( 1st `  x
) )  \  ( Z  X.  Z ) ) 
C_  ( D  \ 
( Z  X.  Z
) ) )  -> 
( ( ( 1st `  ( 1st `  x
) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x
) )  \  ( Z  X.  Z ) ) )  C_  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
9786, 95, 96syl2anc 667 . . . . . . . . 9  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( ( 1st `  ( 1st `  x
) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x
) )  \  ( Z  X.  Z ) ) )  C_  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
98 inundif 3845 . . . . . . . . . 10  |-  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )  =  ( 1st `  ( 1st `  x ) )
9998eqcomi 2460 . . . . . . . . 9  |-  ( 1st `  ( 1st `  x
) )  =  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )
10097, 99, 13sstr4g 3473 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 1st `  ( 1st `  x ) ) 
C_  M )
101 ssid 3451 . . . . . . . . 9  |-  H  C_  H
102101a1i 11 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  H  C_  H )
1037, 8, 45, 46, 47, 49, 100, 102ss2mcls 30206 . . . . . . 7  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( ( 1st `  ( 1st `  x ) ) (mCls `  T ) H )  C_  ( M (mCls `  T ) H ) )
10489, 51eqeltrrd 2530 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  <. ( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  J )
1053, 42, 45elmpps 30211 . . . . . . . . 9  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  J  <->  ( <. ( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T )  /\  A  e.  (
( 1st `  ( 1st `  x ) ) (mCls `  T ) H ) ) )
106105simprbi 466 . . . . . . . 8  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  J  ->  A  e.  ( ( 1st `  ( 1st `  x ) ) (mCls `  T ) H ) )
107104, 106syl 17 . . . . . . 7  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  A  e.  ( ( 1st `  ( 1st `  x
) ) (mCls `  T ) H ) )
108103, 107sseldd 3433 . . . . . 6  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  A  e.  ( M
(mCls `  T ) H ) )
10944, 108rexlimddv 2883 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  ( M
(mCls `  T ) H ) )
1103, 42, 45elmpps 30211 . . . . 5  |-  ( <. M ,  H ,  A >.  e.  J  <->  ( <. M ,  H ,  A >.  e.  (mPreSt `  T
)  /\  A  e.  ( M (mCls `  T
) H ) ) )
11140, 109, 110sylanbrc 670 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. M ,  H ,  A >.  e.  J )
1121ineq1i 3630 . . . . . . . 8  |-  ( M  i^i  ( Z  X.  Z ) )  =  ( ( C  u.  ( D  \  ( Z  X.  Z ) ) )  i^i  ( Z  X.  Z ) )
113 indir 3691 . . . . . . . 8  |-  ( ( C  u.  ( D 
\  ( Z  X.  Z ) ) )  i^i  ( Z  X.  Z ) )  =  ( ( C  i^i  ( Z  X.  Z
) )  u.  (
( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) ) )
114 incom 3625 . . . . . . . . . . 11  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  ( ( Z  X.  Z )  i^i  ( D  \  ( Z  X.  Z ) ) )
115 disjdif 3839 . . . . . . . . . . 11  |-  ( ( Z  X.  Z )  i^i  ( D  \ 
( Z  X.  Z
) ) )  =  (/)
116114, 115eqtri 2473 . . . . . . . . . 10  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  (/)
117 0ss 3763 . . . . . . . . . 10  |-  (/)  C_  ( C  i^i  ( Z  X.  Z ) )
118116, 117eqsstri 3462 . . . . . . . . 9  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  C_  ( C  i^i  ( Z  X.  Z ) )
119 ssequn2 3607 . . . . . . . . 9  |-  ( ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  C_  ( C  i^i  ( Z  X.  Z ) )  <-> 
( ( C  i^i  ( Z  X.  Z
) )  u.  (
( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) ) )  =  ( C  i^i  ( Z  X.  Z
) ) )
120118, 119mpbi 212 . . . . . . . 8  |-  ( ( C  i^i  ( Z  X.  Z ) )  u.  ( ( D 
\  ( Z  X.  Z ) )  i^i  ( Z  X.  Z
) ) )  =  ( C  i^i  ( Z  X.  Z ) )
121112, 113, 1203eqtri 2477 . . . . . . 7  |-  ( M  i^i  ( Z  X.  Z ) )  =  ( C  i^i  ( Z  X.  Z ) )
122121a1i 11 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( M  i^i  ( Z  X.  Z ) )  =  ( C  i^i  ( Z  X.  Z
) ) )
123122oteq1d 4178 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. ( M  i^i  ( Z  X.  Z ) ) ,  H ,  A >.  =  <. ( C  i^i  ( Z  X.  Z
) ) ,  H ,  A >. )
12459, 3, 41, 63msrval 30176 . . . . . 6  |-  ( <. M ,  H ,  A >.  e.  (mPreSt `  T )  ->  ( R `  <. M ,  H ,  A >. )  =  <. ( M  i^i  ( Z  X.  Z
) ) ,  H ,  A >. )
12540, 124syl 17 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  <. ( M  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
126123, 125, 653eqtr4d 2495 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) )
127111, 126jca 535 . . 3  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( <. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) ) )
128127ex 436 . 2  |-  ( T  e. mFS  ->  ( <. C ,  H ,  A >.  e.  U  ->  ( <. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) ) ) )
12941, 42, 2mthmi 30215 . 2  |-  ( (
<. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) )  ->  <. C ,  H ,  A >.  e.  U )
130128, 129impbid1 207 1  |-  ( T  e. mFS  ->  ( <. C ,  H ,  A >.  e.  U  <->  ( <. M ,  H ,  A >.  e.  J  /\  ( R `
 <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   <.cotp 3976   U.cuni 4198    _I cid 4744    X. cxp 4832   `'ccnv 4833   "cima 4837   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   Fincfn 7569  mVRcmvar 30099  mExcmex 30105  mDVcmdv 30106  mVarscmvrs 30107  mPreStcmpst 30111  mStRedcmsr 30112  mFScmfs 30114  mClscmcls 30115  mPPStcmpps 30116  mThmcmthm 30117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-frmd 16633  df-mrex 30124  df-mex 30125  df-mdv 30126  df-mrsub 30128  df-msub 30129  df-mvh 30130  df-mpst 30131  df-msr 30132  df-msta 30133  df-mfs 30134  df-mcls 30135  df-mpps 30136  df-mthm 30137
This theorem is referenced by: (None)
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