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Theorem mthmpps 28919
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 2443 . . . . . . . . . . . . . 14  |-  (mPreSt `  T )  =  (mPreSt `  T )
42, 3mthmsta 28915 . . . . . . . . . . . . 13  |-  U  C_  (mPreSt `  T )
5 simpr 461 . . . . . . . . . . . . 13  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  U )
64, 5sseldi 3487 . . . . . . . . . . . 12  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  (mPreSt `  T ) )
7 mthmpps.d . . . . . . . . . . . . 13  |-  D  =  (mDV `  T )
8 eqid 2443 . . . . . . . . . . . . 13  |-  (mEx `  T )  =  (mEx
`  T )
97, 8, 3elmpst 28873 . . . . . . . . . . . 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 196 . . . . . . . . . . 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 1009 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  C_  D  /\  `' C  =  C
) )
1211simpld 459 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  C  C_  D )
13 difssd 3617 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( D  \  ( Z  X.  Z ) ) 
C_  D )
1412, 13unssd 3665 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  u.  ( D  \  ( Z  X.  Z ) ) ) 
C_  D )
151, 14syl5eqss 3533 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  M  C_  D )
1611simprd 463 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' C  =  C
)
17 cnvdif 5402 . . . . . . . . . . 11  |-  `' ( D  \  ( Z  X.  Z ) )  =  ( `' D  \  `' ( Z  X.  Z ) )
18 cnvdif 5402 . . . . . . . . . . . . . 14  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  ( `' ( (mVR `  T )  X.  (mVR `  T ) )  \  `'  _I  )
19 cnvxp 5414 . . . . . . . . . . . . . . 15  |-  `' ( (mVR `  T )  X.  (mVR `  T )
)  =  ( (mVR
`  T )  X.  (mVR `  T )
)
20 cnvi 5400 . . . . . . . . . . . . . . 15  |-  `'  _I  =  _I
2119, 20difeq12i 3605 . . . . . . . . . . . . . 14  |-  ( `' ( (mVR `  T
)  X.  (mVR `  T ) )  \  `'  _I  )  =  ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
2218, 21eqtri 2472 . . . . . . . . . . . . 13  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  (
( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
23 eqid 2443 . . . . . . . . . . . . . . 15  |-  (mVR `  T )  =  (mVR
`  T )
2423, 7mdvval 28841 . . . . . . . . . . . . . 14  |-  D  =  ( ( (mVR `  T )  X.  (mVR `  T ) )  \  _I  )
2524cnveqi 5167 . . . . . . . . . . . . 13  |-  `' D  =  `' ( ( (mVR
`  T )  X.  (mVR `  T )
)  \  _I  )
2622, 25, 243eqtr4i 2482 . . . . . . . . . . . 12  |-  `' D  =  D
27 cnvxp 5414 . . . . . . . . . . . 12  |-  `' ( Z  X.  Z )  =  ( Z  X.  Z )
2826, 27difeq12i 3605 . . . . . . . . . . 11  |-  ( `' D  \  `' ( Z  X.  Z ) )  =  ( D 
\  ( Z  X.  Z ) )
2917, 28eqtri 2472 . . . . . . . . . 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 3644 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
321cnveqi 5167 . . . . . . . . 9  |-  `' M  =  `' ( C  u.  ( D  \  ( Z  X.  Z ) ) )
33 cnvun 5401 . . . . . . . . 9  |-  `' ( C  u.  ( D 
\  ( Z  X.  Z ) ) )  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3432, 33eqtri 2472 . . . . . . . 8  |-  `' M  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3531, 34, 13eqtr4g 2509 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' M  =  M
)
3615, 35jca 532 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( M  C_  D  /\  `' M  =  M
) )
3710simp2d 1010 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( H  C_  (mEx `  T )  /\  H  e.  Fin ) )
3810simp3d 1011 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  (mEx `  T
) )
397, 8, 3elmpst 28873 . . . . . 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 1181 . . . . 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 28913 . . . . . . 7  |-  ( <. C ,  H ,  A >.  e.  U  <->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
445, 43sylib 196 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
45 eqid 2443 . . . . . . . 8  |-  (mCls `  T )  =  (mCls `  T )
46 simpll 753 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  T  e. mFS )
4715adantr 465 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  M  C_  D )
4837simpld 459 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  H  C_  (mEx `  T
) )
4948adantr 465 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  H  C_  (mEx `  T
) )
503, 42mppspst 28911 . . . . . . . . . . . . . . . . . . 19  |-  J  C_  (mPreSt `  T )
51 simprl 756 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  J )
5250, 51sseldi 3487 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  (mPreSt `  T
) )
533mpst123 28877 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  (mPreSt `  T
)  ->  x  =  <. ( 1st `  ( 1st `  x ) ) ,  ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  x
) >. )
5452, 53syl 16 . . . . . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . 17  |-  U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  =  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )
6159, 3, 41, 60msrval 28875 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . 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 28875 . . . . . . . . . . . . . . . . 17  |-  ( <. C ,  H ,  A >.  e.  (mPreSt `  T )  ->  ( R `  <. C ,  H ,  A >. )  =  <. ( C  i^i  ( Z  X.  Z
) ) ,  H ,  A >. )
656, 64syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. C ,  H ,  A >. )  =  <. ( C  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
6665adantr 465 . . . . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . 16  |-  ( 1st `  ( 1st `  x
) )  e.  _V
6968inex1 4578 . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( 1st `  x
) )  e.  _V
71 fvex 5866 . . . . . . . . . . . . . . 15  |-  ( 2nd `  x )  e.  _V
7269, 70, 71otth 4719 . . . . . . . . . . . . . 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 196 . . . . . . . . . . . . 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 1009 . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  ( 1st `  x ) )  =  H )
7673simp3d 1011 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  x
)  =  A )
7776sneqd 4026 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  { ( 2nd `  x
) }  =  { A } )
7875, 77uneq12d 3644 . . . . . . . . . . . . . . . . 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 5327 . . . . . . . . . . . . . . . 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 4244 . . . . . . . . . . . . . . 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 2502 . . . . . . . . . . . . . 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 5015 . . . . . . . . . . . . 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 3685 . . . . . . . . . . . 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 2486 . . . . . . . . . . 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 3703 . . . . . . . . . . 11  |-  ( C  i^i  ( Z  X.  Z ) )  C_  C
8684, 85syl6eqssr 3540 . . . . . . . . . 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 2444 . . . . . . . . . . . . . . 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 4217 . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . 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 2532 . . . . . . . . . . . 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 28873 . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( ( 1st `  ( 1st `  x
) )  C_  D  /\  `' ( 1st `  ( 1st `  x ) )  =  ( 1st `  ( 1st `  x ) ) ) )
9392simpld 459 . . . . . . . . . . . 12  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( 1st `  ( 1st `  x
) )  C_  D
)
9490, 93syl 16 . . . . . . . . . . 11  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 1st `  ( 1st `  x ) ) 
C_  D )
9594ssdifd 3625 . . . . . . . . . 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 3661 . . . . . . . . . 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 661 . . . . . . . . 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 3892 . . . . . . . . . 10  |-  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )  =  ( 1st `  ( 1st `  x ) )
9998eqcomi 2456 . . . . . . . . 9  |-  ( 1st `  ( 1st `  x
) )  =  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )
10097, 99, 13sstr4g 3530 . . . . . . . 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 3508 . . . . . . . . 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 28905 . . . . . . 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 2532 . . . . . . . 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 28910 . . . . . . . . 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 464 . . . . . . . 8  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  J  ->  A  e.  ( ( 1st `  ( 1st `  x ) ) (mCls `  T ) H ) )
107104, 106syl 16 . . . . . . 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 3490 . . . . . 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 2939 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  ( M
(mCls `  T ) H ) )
1103, 42, 45elmpps 28910 . . . . 5  |-  ( <. M ,  H ,  A >.  e.  J  <->  ( <. M ,  H ,  A >.  e.  (mPreSt `  T
)  /\  A  e.  ( M (mCls `  T
) H ) ) )
11140, 109, 110sylanbrc 664 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. M ,  H ,  A >.  e.  J )
1121ineq1i 3681 . . . . . . . 8  |-  ( M  i^i  ( Z  X.  Z ) )  =  ( ( C  u.  ( D  \  ( Z  X.  Z ) ) )  i^i  ( Z  X.  Z ) )
113 indir 3731 . . . . . . . 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 3676 . . . . . . . . . . 11  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  ( ( Z  X.  Z )  i^i  ( D  \  ( Z  X.  Z ) ) )
115 disjdif 3886 . . . . . . . . . . 11  |-  ( ( Z  X.  Z )  i^i  ( D  \ 
( Z  X.  Z
) ) )  =  (/)
116114, 115eqtri 2472 . . . . . . . . . 10  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  (/)
117 0ss 3800 . . . . . . . . . 10  |-  (/)  C_  ( C  i^i  ( Z  X.  Z ) )
118116, 117eqsstri 3519 . . . . . . . . 9  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  C_  ( C  i^i  ( Z  X.  Z ) )
119 ssequn2 3662 . . . . . . . . 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 208 . . . . . . . 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 2476 . . . . . . 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 4214 . . . . 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 28875 . . . . . 6  |-  ( <. M ,  H ,  A >.  e.  (mPreSt `  T )  ->  ( R `  <. M ,  H ,  A >. )  =  <. ( M  i^i  ( Z  X.  Z
) ) ,  H ,  A >. )
12540, 124syl 16 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  <. ( M  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
126123, 125, 653eqtr4d 2494 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) )
127111, 126jca 532 . . 3  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( <. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) ) )
128127ex 434 . 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 28914 . 2  |-  ( (
<. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) )  ->  <. C ,  H ,  A >.  e.  U )
130128, 129impbid1 203 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 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E.wrex 2794    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   <.cotp 4022   U.cuni 4234    _I cid 4780    X. cxp 4987   `'ccnv 4988   "cima 4992   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   Fincfn 7518  mVRcmvar 28798  mExcmex 28804  mDVcmdv 28805  mVarscmvrs 28806  mPreStcmpst 28810  mStRedcmsr 28811  mFScmfs 28813  mClscmcls 28814  mPPStcmpps 28815  mThmcmthm 28816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11092  df-fz 11683  df-fzo 11806  df-seq 12089  df-hash 12387  df-word 12523  df-concat 12525  df-s1 12526  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-0g 14820  df-gsum 14821  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-submnd 15945  df-frmd 15995  df-mrex 28823  df-mex 28824  df-mdv 28825  df-mrsub 28827  df-msub 28828  df-mvh 28829  df-mpst 28830  df-msr 28831  df-msta 28832  df-mfs 28833  df-mcls 28834  df-mpps 28835  df-mthm 28836
This theorem is referenced by: (None)
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