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Theorem mthmpps 30292
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 2471 . . . . . . . . . . . . . 14  |-  (mPreSt `  T )  =  (mPreSt `  T )
42, 3mthmsta 30288 . . . . . . . . . . . . 13  |-  U  C_  (mPreSt `  T )
5 simpr 468 . . . . . . . . . . . . 13  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  U )
64, 5sseldi 3416 . . . . . . . . . . . 12  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. C ,  H ,  A >.  e.  (mPreSt `  T ) )
7 mthmpps.d . . . . . . . . . . . . 13  |-  D  =  (mDV `  T )
8 eqid 2471 . . . . . . . . . . . . 13  |-  (mEx `  T )  =  (mEx
`  T )
97, 8, 3elmpst 30246 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 1042 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  C_  D  /\  `' C  =  C
) )
1211simpld 466 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  C  C_  D )
13 difssd 3550 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( D  \  ( Z  X.  Z ) ) 
C_  D )
1412, 13unssd 3601 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  u.  ( D  \  ( Z  X.  Z ) ) ) 
C_  D )
151, 14syl5eqss 3462 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  M  C_  D )
1611simprd 470 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' C  =  C
)
17 cnvdif 5248 . . . . . . . . . . 11  |-  `' ( D  \  ( Z  X.  Z ) )  =  ( `' D  \  `' ( Z  X.  Z ) )
18 cnvdif 5248 . . . . . . . . . . . . . 14  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  ( `' ( (mVR `  T )  X.  (mVR `  T ) )  \  `'  _I  )
19 cnvxp 5260 . . . . . . . . . . . . . . 15  |-  `' ( (mVR `  T )  X.  (mVR `  T )
)  =  ( (mVR
`  T )  X.  (mVR `  T )
)
20 cnvi 5246 . . . . . . . . . . . . . . 15  |-  `'  _I  =  _I
2119, 20difeq12i 3538 . . . . . . . . . . . . . 14  |-  ( `' ( (mVR `  T
)  X.  (mVR `  T ) )  \  `'  _I  )  =  ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
2218, 21eqtri 2493 . . . . . . . . . . . . 13  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  (
( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
23 eqid 2471 . . . . . . . . . . . . . . 15  |-  (mVR `  T )  =  (mVR
`  T )
2423, 7mdvval 30214 . . . . . . . . . . . . . 14  |-  D  =  ( ( (mVR `  T )  X.  (mVR `  T ) )  \  _I  )
2524cnveqi 5014 . . . . . . . . . . . . 13  |-  `' D  =  `' ( ( (mVR
`  T )  X.  (mVR `  T )
)  \  _I  )
2622, 25, 243eqtr4i 2503 . . . . . . . . . . . 12  |-  `' D  =  D
27 cnvxp 5260 . . . . . . . . . . . 12  |-  `' ( Z  X.  Z )  =  ( Z  X.  Z )
2826, 27difeq12i 3538 . . . . . . . . . . 11  |-  ( `' D  \  `' ( Z  X.  Z ) )  =  ( D 
\  ( Z  X.  Z ) )
2917, 28eqtri 2493 . . . . . . . . . 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 3580 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
321cnveqi 5014 . . . . . . . . 9  |-  `' M  =  `' ( C  u.  ( D  \  ( Z  X.  Z ) ) )
33 cnvun 5247 . . . . . . . . 9  |-  `' ( C  u.  ( D 
\  ( Z  X.  Z ) ) )  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3432, 33eqtri 2493 . . . . . . . 8  |-  `' M  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3531, 34, 13eqtr4g 2530 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' M  =  M
)
3615, 35jca 541 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( M  C_  D  /\  `' M  =  M
) )
3710simp2d 1043 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( H  C_  (mEx `  T )  /\  H  e.  Fin ) )
3810simp3d 1044 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  (mEx `  T
) )
397, 8, 3elmpst 30246 . . . . . 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 1214 . . . . 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 30286 . . . . . . 7  |-  ( <. C ,  H ,  A >.  e.  U  <->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
445, 43sylib 201 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  E. x  e.  J  ( R `  x )  =  ( R `  <. C ,  H ,  A >. ) )
45 eqid 2471 . . . . . . . 8  |-  (mCls `  T )  =  (mCls `  T )
46 simpll 768 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  T  e. mFS )
4715adantr 472 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  M  C_  D )
4837simpld 466 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  H  C_  (mEx `  T
) )
4948adantr 472 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  H  C_  (mEx `  T
) )
503, 42mppspst 30284 . . . . . . . . . . . . . . . . . . 19  |-  J  C_  (mPreSt `  T )
51 simprl 772 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  J )
5250, 51sseldi 3416 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  x  e.  (mPreSt `  T
) )
533mpst123 30250 . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . . . . 17  |-  U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  =  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )
6159, 3, 41, 60msrval 30248 . . . . . . . . . . . . . . . 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 30248 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . . . . . 16  |-  ( 1st `  ( 1st `  x
) )  e.  _V
6968inex1 4537 . . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( 1st `  x
) )  e.  _V
71 fvex 5889 . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . 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 1042 . . . . . . . . . . . 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 1043 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  ( 1st `  x ) )  =  H )
7673simp3d 1044 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  x
)  =  A )
7776sneqd 3971 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  { ( 2nd `  x
) }  =  { A } )
7875, 77uneq12d 3580 . . . . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . . . . . 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 4200 . . . . . . . . . . . . . . 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 2523 . . . . . . . . . . . . . 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 4865 . . . . . . . . . . . . 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 3625 . . . . . . . . . . . 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 2507 . . . . . . . . . . 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 3643 . . . . . . . . . . 11  |-  ( C  i^i  ( Z  X.  Z ) )  C_  C
8684, 85syl6eqssr 3469 . . . . . . . . . 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 2472 . . . . . . . . . . . . . . 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 4173 . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . 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 2550 . . . . . . . . . . . 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 30246 . . . . . . . . . . . . . 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 1045 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( ( 1st `  ( 1st `  x
) )  C_  D  /\  `' ( 1st `  ( 1st `  x ) )  =  ( 1st `  ( 1st `  x ) ) ) )
9392simpld 466 . . . . . . . . . . . 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 3558 . . . . . . . . . 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 3597 . . . . . . . . . 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 673 . . . . . . . . 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 3836 . . . . . . . . . 10  |-  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )  =  ( 1st `  ( 1st `  x ) )
9998eqcomi 2480 . . . . . . . . 9  |-  ( 1st `  ( 1st `  x
) )  =  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )
10097, 99, 13sstr4g 3459 . . . . . . . 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 3437 . . . . . . . . 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 30278 . . . . . . 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 2550 . . . . . . . 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 30283 . . . . . . . . 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 471 . . . . . . . 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 3419 . . . . . 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 2875 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  ( M
(mCls `  T ) H ) )
1103, 42, 45elmpps 30283 . . . . 5  |-  ( <. M ,  H ,  A >.  e.  J  <->  ( <. M ,  H ,  A >.  e.  (mPreSt `  T
)  /\  A  e.  ( M (mCls `  T
) H ) ) )
11140, 109, 110sylanbrc 677 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. M ,  H ,  A >.  e.  J )
1121ineq1i 3621 . . . . . . . 8  |-  ( M  i^i  ( Z  X.  Z ) )  =  ( ( C  u.  ( D  \  ( Z  X.  Z ) ) )  i^i  ( Z  X.  Z ) )
113 indir 3682 . . . . . . . 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 3616 . . . . . . . . . . 11  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  ( ( Z  X.  Z )  i^i  ( D  \  ( Z  X.  Z ) ) )
115 disjdif 3830 . . . . . . . . . . 11  |-  ( ( Z  X.  Z )  i^i  ( D  \ 
( Z  X.  Z
) ) )  =  (/)
116114, 115eqtri 2493 . . . . . . . . . 10  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  (/)
117 0ss 3766 . . . . . . . . . 10  |-  (/)  C_  ( C  i^i  ( Z  X.  Z ) )
118116, 117eqsstri 3448 . . . . . . . . 9  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  C_  ( C  i^i  ( Z  X.  Z ) )
119 ssequn2 3598 . . . . . . . . 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 213 . . . . . . . 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 2497 . . . . . . 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 4170 . . . . 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 30248 . . . . . 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 2515 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) )
127111, 126jca 541 . . 3  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( <. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) ) )
128127ex 441 . 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 30287 . 2  |-  ( (
<. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `  <. C ,  H ,  A >. ) )  ->  <. C ,  H ,  A >.  e.  U )
130128, 129impbid1 208 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   <.cotp 3967   U.cuni 4190    _I cid 4749    X. cxp 4837   `'ccnv 4838   "cima 4842   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   Fincfn 7587  mVRcmvar 30171  mExcmex 30177  mDVcmdv 30178  mVarscmvrs 30179  mPreStcmpst 30183  mStRedcmsr 30184  mFScmfs 30186  mClscmcls 30187  mPPStcmpps 30188  mThmcmthm 30189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-frmd 16711  df-mrex 30196  df-mex 30197  df-mdv 30198  df-mrsub 30200  df-msub 30201  df-mvh 30202  df-mpst 30203  df-msr 30204  df-msta 30205  df-mfs 30206  df-mcls 30207  df-mpps 30208  df-mthm 30209
This theorem is referenced by: (None)
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