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Theorem mthmpps 30008
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 2429 . . . . . . . . . . . . . 14  |-  (mPreSt `  T )  =  (mPreSt `  T )
42, 3mthmsta 30004 . . . . . . . . . . . . 13  |-  U  C_  (mPreSt `  T )
5 simpr 462 . . . . . . . . . . . . 13  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  U )
64, 5sseldi 3468 . . . . . . . . . . . 12  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  (mPreSt `  T ) )
7 mthmpps.d . . . . . . . . . . . . 13  |-  D  =  (mDV `  T )
8 eqid 2429 . . . . . . . . . . . . 13  |-  (mEx `  T )  =  (mEx
`  T )
97, 8, 3elmpst 29962 . . . . . . . . . . . 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 199 . . . . . . . . . . 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 1017 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  C_  D  /\  `' C  =  C
) )
1211simpld 460 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  C  C_  D )
13 difssd 3599 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( D  \  ( Z  X.  Z ) ) 
C_  D )
1412, 13unssd 3648 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  u.  ( D  \  ( Z  X.  Z ) ) ) 
C_  D )
151, 14syl5eqss 3514 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  M  C_  D )
1611simprd 464 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' C  =  C
)
17 cnvdif 5262 . . . . . . . . . . 11  |-  `' ( D  \  ( Z  X.  Z ) )  =  ( `' D  \  `' ( Z  X.  Z ) )
18 cnvdif 5262 . . . . . . . . . . . . . 14  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  ( `' ( (mVR `  T )  X.  (mVR `  T ) )  \  `'  _I  )
19 cnvxp 5274 . . . . . . . . . . . . . . 15  |-  `' ( (mVR `  T )  X.  (mVR `  T )
)  =  ( (mVR
`  T )  X.  (mVR `  T )
)
20 cnvi 5260 . . . . . . . . . . . . . . 15  |-  `'  _I  =  _I
2119, 20difeq12i 3587 . . . . . . . . . . . . . 14  |-  ( `' ( (mVR `  T
)  X.  (mVR `  T ) )  \  `'  _I  )  =  ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
2218, 21eqtri 2458 . . . . . . . . . . . . 13  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  (
( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
23 eqid 2429 . . . . . . . . . . . . . . 15  |-  (mVR `  T )  =  (mVR
`  T )
2423, 7mdvval 29930 . . . . . . . . . . . . . 14  |-  D  =  ( ( (mVR `  T )  X.  (mVR `  T ) )  \  _I  )
2524cnveqi 5029 . . . . . . . . . . . . 13  |-  `' D  =  `' ( ( (mVR
`  T )  X.  (mVR `  T )
)  \  _I  )
2622, 25, 243eqtr4i 2468 . . . . . . . . . . . 12  |-  `' D  =  D
27 cnvxp 5274 . . . . . . . . . . . 12  |-  `' ( Z  X.  Z )  =  ( Z  X.  Z )
2826, 27difeq12i 3587 . . . . . . . . . . 11  |-  ( `' D  \  `' ( Z  X.  Z ) )  =  ( D 
\  ( Z  X.  Z ) )
2917, 28eqtri 2458 . . . . . . . . . 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 3627 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
321cnveqi 5029 . . . . . . . . 9  |-  `' M  =  `' ( C  u.  ( D  \  ( Z  X.  Z ) ) )
33 cnvun 5261 . . . . . . . . 9  |-  `' ( C  u.  ( D 
\  ( Z  X.  Z ) ) )  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3432, 33eqtri 2458 . . . . . . . 8  |-  `' M  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3531, 34, 13eqtr4g 2495 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' M  =  M
)
3615, 35jca 534 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( M  C_  D  /\  `' M  =  M
) )
3710simp2d 1018 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( H  C_  (mEx `  T )  /\  H  e.  Fin ) )
3810simp3d 1019 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  (mEx `  T
) )
397, 8, 3elmpst 29962 . . . . . 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 1189 . . . . 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 30002 . . . . . . 7  |-  ( <. C ,  H ,  A >.  e.  U  <->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
445, 43sylib 199 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
45 eqid 2429 . . . . . . . 8  |-  (mCls `  T )  =  (mCls `  T )
46 simpll 758 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  T  e. mFS )
4715adantr 466 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  M  C_  D )
4837simpld 460 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  H  C_  (mEx `  T
) )
4948adantr 466 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  H  C_  (mEx `  T
) )
503, 42mppspst 30000 . . . . . . . . . . . . . . . . . . 19  |-  J  C_  (mPreSt `  T )
51 simprl 762 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  J )
5250, 51sseldi 3468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  (mPreSt `  T
) )
533mpst123 29966 . . . . . . . . . . . . . . . . . 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 5885 . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . 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 2518 . . . . . . . . . . . . . . . 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 2429 . . . . . . . . . . . . . . . . 17  |-  U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  =  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )
6159, 3, 41, 60msrval 29964 . . . . . . . . . . . . . . . 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 29964 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . 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 5891 . . . . . . . . . . . . . . . 16  |-  ( 1st `  ( 1st `  x
) )  e.  _V
6968inex1 4566 . . . . . . . . . . . . . . 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 5891 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( 1st `  x
) )  e.  _V
71 fvex 5891 . . . . . . . . . . . . . . 15  |-  ( 2nd `  x )  e.  _V
7269, 70, 71otth 4704 . . . . . . . . . . . . . 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 199 . . . . . . . . . . . . 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 1017 . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  ( 1st `  x ) )  =  H )
7673simp3d 1019 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  x
)  =  A )
7776sneqd 4014 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  { ( 2nd `  x
) }  =  { A } )
7875, 77uneq12d 3627 . . . . . . . . . . . . . . . . 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 5188 . . . . . . . . . . . . . . . 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 4232 . . . . . . . . . . . . . . 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 2488 . . . . . . . . . . . . . 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 4880 . . . . . . . . . . . . 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 3670 . . . . . . . . . . . 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 2472 . . . . . . . . . . 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 3688 . . . . . . . . . . 11  |-  ( C  i^i  ( Z  X.  Z ) )  C_  C
8684, 85syl6eqssr 3521 . . . . . . . . . 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 2430 . . . . . . . . . . . . . . 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 4205 . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . 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 2518 . . . . . . . . . . . 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 29962 . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( ( 1st `  ( 1st `  x
) )  C_  D  /\  `' ( 1st `  ( 1st `  x ) )  =  ( 1st `  ( 1st `  x ) ) ) )
9392simpld 460 . . . . . . . . . . . 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 3607 . . . . . . . . . 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 3644 . . . . . . . . . 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 665 . . . . . . . . 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 3879 . . . . . . . . . 10  |-  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )  =  ( 1st `  ( 1st `  x ) )
9998eqcomi 2442 . . . . . . . . 9  |-  ( 1st `  ( 1st `  x
) )  =  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )
10097, 99, 13sstr4g 3511 . . . . . . . 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 3489 . . . . . . . . 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 29994 . . . . . . 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 2518 . . . . . . . 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 29999 . . . . . . . . 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 465 . . . . . . . 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 3471 . . . . . 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 2928 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  ( M
(mCls `  T ) H ) )
1103, 42, 45elmpps 29999 . . . . 5  |-  ( <. M ,  H ,  A >.  e.  J  <->  ( <. M ,  H ,  A >.  e.  (mPreSt `  T
)  /\  A  e.  ( M (mCls `  T
) H ) ) )
11140, 109, 110sylanbrc 668 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. M ,  H ,  A >.  e.  J )
1121ineq1i 3666 . . . . . . . 8  |-  ( M  i^i  ( Z  X.  Z ) )  =  ( ( C  u.  ( D  \  ( Z  X.  Z ) ) )  i^i  ( Z  X.  Z ) )
113 indir 3727 . . . . . . . 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 3661 . . . . . . . . . . 11  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  ( ( Z  X.  Z )  i^i  ( D  \  ( Z  X.  Z ) ) )
115 disjdif 3873 . . . . . . . . . . 11  |-  ( ( Z  X.  Z )  i^i  ( D  \ 
( Z  X.  Z
) ) )  =  (/)
116114, 115eqtri 2458 . . . . . . . . . 10  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  (/)
117 0ss 3797 . . . . . . . . . 10  |-  (/)  C_  ( C  i^i  ( Z  X.  Z ) )
118116, 117eqsstri 3500 . . . . . . . . 9  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  C_  ( C  i^i  ( Z  X.  Z ) )
119 ssequn2 3645 . . . . . . . . 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 211 . . . . . . . 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 2462 . . . . . . 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 4202 . . . . 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 29964 . . . . . 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 2480 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) )
127111, 126jca 534 . . 3  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( <. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) ) )
128127ex 435 . 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 30003 . 2  |-  ( (
<. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) )  ->  <. C ,  H ,  A >.  e.  U )
130128, 129impbid1 206 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   <.cotp 4010   U.cuni 4222    _I cid 4764    X. cxp 4852   `'ccnv 4853   "cima 4857   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Fincfn 7577  mVRcmvar 29887  mExcmex 29893  mDVcmdv 29894  mVarscmvrs 29895  mPreStcmpst 29899  mStRedcmsr 29900  mFScmfs 29902  mClscmcls 29903  mPPStcmpps 29904  mThmcmthm 29905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-word 12651  df-concat 12653  df-s1 12654  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-gsum 15300  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-frmd 16584  df-mrex 29912  df-mex 29913  df-mdv 29914  df-mrsub 29916  df-msub 29917  df-mvh 29918  df-mpst 29919  df-msr 29920  df-msta 29921  df-mfs 29922  df-mcls 29923  df-mpps 29924  df-mthm 29925
This theorem is referenced by: (None)
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