Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj944 Structured version   Unicode version

Theorem bnj944 31765
Description: Technical lemma for bnj69 31835. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj944.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj944.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj944.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj944.4  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj944.7  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj944.10  |-  D  =  ( om  \  { (/)
} )
bnj944.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj944.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj944.14  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj944.15  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
Assertion
Ref Expression
bnj944  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
Distinct variable groups:    A, f,
i, m, n    y, A, f, i, m    R, f, i, m, n    y, R    f, X, n
Allowed substitution hints:    ph( y, f, i, m, n, p)    ps( y, f, i, m, n, p)    ch( y,
f, i, m, n, p)    ta( y, f, i, m, n, p)    si( y,
f, i, m, n, p)    A( p)    C( y,
f, i, m, n, p)    D( y, f, i, m, n, p)    R( p)    G( y, f, i, m, n, p)    X( y, i, m, p)    ph'( y, f, i, m, n, p)    ph"( y, f, i, m, n, p)

Proof of Theorem bnj944
StepHypRef Expression
1 simpl 454 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A
) )
2 bnj944.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
3 bnj667 31578 . . . . . . . 8  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )
42, 3sylbi 195 . . . . . . 7  |-  ( ch 
->  ( f  Fn  n  /\  ph  /\  ps )
)
5 bnj944.14 . . . . . . 7  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
64, 5sylibr 212 . . . . . 6  |-  ( ch 
->  ta )
763ad2ant1 1004 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ta )
87adantl 463 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ta )
92bnj1232 31631 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
10 vex 2973 . . . . . . . 8  |-  m  e. 
_V
1110bnj216 31557 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
12 id 22 . . . . . . 7  |-  ( p  =  suc  n  ->  p  =  suc  n )
139, 11, 123anim123i 1168 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
14 bnj944.15 . . . . . . 7  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
15 3ancomb 969 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1614, 15bitri 249 . . . . . 6  |-  ( si  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1713, 16sylibr 212 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  si )
1817adantl 463 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  si )
19 bnj253 31526 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ta  /\ 
si ) )
201, 8, 18, 19syl3anbrc 1167 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )
)
21 bnj944.12 . . . 4  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
22 bnj944.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
23 bnj944.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
24 bnj944.2 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2522, 5, 14, 23, 24bnj938 31764 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2525 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  C  e.  _V )
2720, 26syl 16 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
28 bnj658 31577 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
n  e.  D  /\  f  Fn  n  /\  ph ) )
292, 28sylbi 195 . . . . 5  |-  ( ch 
->  ( n  e.  D  /\  f  Fn  n  /\  ph ) )
30293ad2ant1 1004 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  f  Fn  n  /\  ph ) )
31 simp3 985 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  p  =  suc  n )
32 bnj291 31533 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  p  =  suc  n ) )
3330, 31, 32sylanbrc 659 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph ) )
3433adantl 463 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph ) )
35 bnj944.7 . . . . 5  |-  ( ph"  <->  [. G  / 
f ]. ph' )
36 bnj944.13 . . . . . . 7  |-  G  =  ( f  u.  { <. n ,  C >. } )
37 opeq2 4057 . . . . . . . . 9  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
3837sneqd 3886 . . . . . . . 8  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
3938uneq2d 3507 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
4036, 39syl5eq 2485 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
41 dfsbcq 3185 . . . . . 6  |-  ( G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  ->  ( [. G  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4240, 41syl 16 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( [. G  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4335, 42syl5bb 257 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( ph"  <->  [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4443imbi2d 316 . . 3  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )  <->  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) ) )
45 bnj944.4 . . . 4  |-  ( ph'  <->  [. p  /  n ]. ph )
46 biid 236 . . . 4  |-  ( [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' )
47 eqid 2441 . . . 4  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
48 0ex 4419 . . . . 5  |-  (/)  e.  _V
4948elimel 3849 . . . 4  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
5023, 45, 46, 22, 47, 49bnj929 31763 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  [. ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  / 
f ]. ph' )
5144, 50dedth 3838 . 2  |-  ( C  e.  _V  ->  (
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" ) )
5227, 34, 51sylc 60 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970   [.wsbc 3183    \ cdif 3322    u. cun 3323   (/)c0 3634   ifcif 3788   {csn 3874   <.cop 3880   U_ciun 4168   suc csuc 4717    Fn wfn 5410   ` cfv 5415   omcom 6475    /\ w-bnj17 31508    predc-bnj14 31510    FrSe w-bnj15 31514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371  ax-reg 7803
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-bnj17 31509  df-bnj14 31511  df-bnj13 31513  df-bnj15 31515
This theorem is referenced by:  bnj910  31775
  Copyright terms: Public domain W3C validator