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Theorem mthmpps 29209
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 2454 . . . . . . . . . . . . . 14  |-  (mPreSt `  T )  =  (mPreSt `  T )
42, 3mthmsta 29205 . . . . . . . . . . . . 13  |-  U  C_  (mPreSt `  T )
5 simpr 459 . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . 13  |-  (mEx `  T )  =  (mEx
`  T )
97, 8, 3elmpst 29163 . . . . . . . . . . . 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 1006 . . . . . . . . . 10  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( C  C_  D  /\  `' C  =  C
) )
1211simpld 457 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  C  C_  D )
13 difssd 3618 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( D  \  ( Z  X.  Z ) ) 
C_  D )
1412, 13unssd 3666 . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' C  =  C
)
17 cnvdif 5397 . . . . . . . . . . 11  |-  `' ( D  \  ( Z  X.  Z ) )  =  ( `' D  \  `' ( Z  X.  Z ) )
18 cnvdif 5397 . . . . . . . . . . . . . 14  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  ( `' ( (mVR `  T )  X.  (mVR `  T ) )  \  `'  _I  )
19 cnvxp 5409 . . . . . . . . . . . . . . 15  |-  `' ( (mVR `  T )  X.  (mVR `  T )
)  =  ( (mVR
`  T )  X.  (mVR `  T )
)
20 cnvi 5395 . . . . . . . . . . . . . . 15  |-  `'  _I  =  _I
2119, 20difeq12i 3606 . . . . . . . . . . . . . 14  |-  ( `' ( (mVR `  T
)  X.  (mVR `  T ) )  \  `'  _I  )  =  ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
2218, 21eqtri 2483 . . . . . . . . . . . . 13  |-  `' ( ( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )  =  (
( (mVR `  T
)  X.  (mVR `  T ) )  \  _I  )
23 eqid 2454 . . . . . . . . . . . . . . 15  |-  (mVR `  T )  =  (mVR
`  T )
2423, 7mdvval 29131 . . . . . . . . . . . . . 14  |-  D  =  ( ( (mVR `  T )  X.  (mVR `  T ) )  \  _I  )
2524cnveqi 5166 . . . . . . . . . . . . 13  |-  `' D  =  `' ( ( (mVR
`  T )  X.  (mVR `  T )
)  \  _I  )
2622, 25, 243eqtr4i 2493 . . . . . . . . . . . 12  |-  `' D  =  D
27 cnvxp 5409 . . . . . . . . . . . 12  |-  `' ( Z  X.  Z )  =  ( Z  X.  Z )
2826, 27difeq12i 3606 . . . . . . . . . . 11  |-  ( `' D  \  `' ( Z  X.  Z ) )  =  ( D 
\  ( Z  X.  Z ) )
2917, 28eqtri 2483 . . . . . . . . . 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 3645 . . . . . . . 8  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )  =  ( C  u.  ( D  \  ( Z  X.  Z ) ) ) )
321cnveqi 5166 . . . . . . . . 9  |-  `' M  =  `' ( C  u.  ( D  \  ( Z  X.  Z ) ) )
33 cnvun 5396 . . . . . . . . 9  |-  `' ( C  u.  ( D 
\  ( Z  X.  Z ) ) )  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3432, 33eqtri 2483 . . . . . . . 8  |-  `' M  =  ( `' C  u.  `' ( D  \ 
( Z  X.  Z
) ) )
3531, 34, 13eqtr4g 2520 . . . . . . 7  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  `' M  =  M
)
3615, 35jca 530 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( M  C_  D  /\  `' M  =  M
) )
3710simp2d 1007 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( H  C_  (mEx `  T )  /\  H  e.  Fin ) )
3810simp3d 1008 . . . . . 6  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  (mEx `  T
) )
397, 8, 3elmpst 29163 . . . . . 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 1178 . . . . 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 29203 . . . . . . 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 2454 . . . . . . . 8  |-  (mCls `  T )  =  (mCls `  T )
46 simpll 751 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  T  e. mFS )
4715adantr 463 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  M  C_  D )
4837simpld 457 . . . . . . . . 9  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  H  C_  (mEx `  T
) )
4948adantr 463 . . . . . . . 8  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  H  C_  (mEx `  T
) )
503, 42mppspst 29201 . . . . . . . . . . . . . . . . . . 19  |-  J  C_  (mPreSt `  T )
51 simprl 754 . . . . . . . . . . . . . . . . . . 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 29167 . . . . . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . . . 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 2543 . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . 17  |-  U. ( V " ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )  =  U. ( V
" ( ( 2nd `  ( 1st `  x
) )  u.  {
( 2nd `  x
) } ) )
6159, 3, 41, 60msrval 29165 . . . . . . . . . . . . . . . 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 29165 . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 2503 . . . . . . . . . . . . . 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 5858 . . . . . . . . . . . . . . . 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 5858 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( 1st `  x
) )  e.  _V
71 fvex 5858 . . . . . . . . . . . . . . 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 1006 . . . . . . . . . . . 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 1007 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  ( 1st `  x ) )  =  H )
7673simp3d 1008 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  -> 
( 2nd `  x
)  =  A )
7776sneqd 4028 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  /\  ( x  e.  J  /\  ( R `
 x )  =  ( R `  <. C ,  H ,  A >. ) ) )  ->  { ( 2nd `  x
) }  =  { A } )
7875, 77uneq12d 3645 . . . . . . . . . . . . . . . . 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 5325 . . . . . . . . . . . . . . . 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 4245 . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . 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 5014 . . . . . . . . . . . . 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 3686 . . . . . . . . . . . 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 2497 . . . . . . . . . . 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 3704 . . . . . . . . . . 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 2455 . . . . . . . . . . . . . . 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 4218 . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . 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 2543 . . . . . . . . . . . 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 29163 . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  H ,  A >.  e.  (mPreSt `  T
)  ->  ( ( 1st `  ( 1st `  x
) )  C_  D  /\  `' ( 1st `  ( 1st `  x ) )  =  ( 1st `  ( 1st `  x ) ) ) )
9392simpld 457 . . . . . . . . . . . 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 3626 . . . . . . . . . 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 3662 . . . . . . . . . 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 659 . . . . . . . . 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 3894 . . . . . . . . . 10  |-  ( ( ( 1st `  ( 1st `  x ) )  i^i  ( Z  X.  Z ) )  u.  ( ( 1st `  ( 1st `  x ) ) 
\  ( Z  X.  Z ) ) )  =  ( 1st `  ( 1st `  x ) )
9998eqcomi 2467 . . . . . . . . 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 29195 . . . . . . 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 2543 . . . . . . . 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 29200 . . . . . . . . 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 462 . . . . . . . 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 2950 . . . . 5  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  A  e.  ( M
(mCls `  T ) H ) )
1103, 42, 45elmpps 29200 . . . . 5  |-  ( <. M ,  H ,  A >.  e.  J  <->  ( <. M ,  H ,  A >.  e.  (mPreSt `  T
)  /\  A  e.  ( M (mCls `  T
) H ) ) )
11140, 109, 110sylanbrc 662 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  ->  <. M ,  H ,  A >.  e.  J )
1121ineq1i 3682 . . . . . . . 8  |-  ( M  i^i  ( Z  X.  Z ) )  =  ( ( C  u.  ( D  \  ( Z  X.  Z ) ) )  i^i  ( Z  X.  Z ) )
113 indir 3743 . . . . . . . 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 3677 . . . . . . . . . . 11  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  ( ( Z  X.  Z )  i^i  ( D  \  ( Z  X.  Z ) ) )
115 disjdif 3888 . . . . . . . . . . 11  |-  ( ( Z  X.  Z )  i^i  ( D  \ 
( Z  X.  Z
) ) )  =  (/)
116114, 115eqtri 2483 . . . . . . . . . 10  |-  ( ( D  \  ( Z  X.  Z ) )  i^i  ( Z  X.  Z ) )  =  (/)
117 0ss 3813 . . . . . . . . . 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 3663 . . . . . . . . 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 2487 . . . . . . 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 4215 . . . . 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 29165 . . . . . 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 2505 . . . 4  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) )
127111, 126jca 530 . . 3  |-  ( ( T  e. mFS  /\  <. C ,  H ,  A >.  e.  U )  -> 
( <. M ,  H ,  A >.  e.  J  /\  ( R `  <. M ,  H ,  A >. )  =  ( R `
 <. C ,  H ,  A >. ) ) )
128127ex 432 . 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 29204 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   <.cotp 4024   U.cuni 4235    _I cid 4779    X. cxp 4986   `'ccnv 4987   "cima 4991   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Fincfn 7509  mVRcmvar 29088  mExcmex 29094  mDVcmdv 29095  mVarscmvrs 29096  mPreStcmpst 29100  mStRedcmsr 29101  mFScmfs 29103  mClscmcls 29104  mPPStcmpps 29105  mThmcmthm 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-frmd 16219  df-mrex 29113  df-mex 29114  df-mdv 29115  df-mrsub 29117  df-msub 29118  df-mvh 29119  df-mpst 29120  df-msr 29121  df-msta 29122  df-mfs 29123  df-mcls 29124  df-mpps 29125  df-mthm 29126
This theorem is referenced by: (None)
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