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Theorem bnj998 32970
Description: Technical lemma for bnj69 33022. 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
bnj998.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj998.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj998.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj998.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
bnj998.5  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
bnj998.7  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj998.8  |-  ( ps'  <->  [. p  /  n ]. ps )
bnj998.9  |-  ( ch'  <->  [. p  /  n ]. ch )
bnj998.10  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj998.11  |-  ( ps"  <->  [. G  / 
f ]. ps' )
bnj998.12  |-  ( ch"  <->  [. G  / 
f ]. ch' )
bnj998.13  |-  D  =  ( om  \  { (/)
} )
bnj998.14  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj998.15  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj998.16  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj998  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
Distinct variable groups:    A, f,
i, m, n, y    D, f, i, n    i, G    R, f, i, m, n, y    f, X, i, n    f, p, i, n    ph, i
Allowed substitution hints:    ph( y, z, f, m, n, p)    ps( y, z, f, i, m, n, p)    ch( y, z, f, i, m, n, p)    th( y,
z, f, i, m, n, p)    ta( y,
z, f, i, m, n, p)    et( y,
z, f, i, m, n, p)    A( z, p)    B( y, z, f, i, m, n, p)    C( y, z, f, i, m, n, p)    D( y, z, m, p)    R( z, p)    G( y, z, f, m, n, p)    X( y, z, m, p)    ph'( y, z, f, i, m, n, p)    ps'( y, z, f, i, m, n, p)    ch'( y, z, f, i, m, n, p)    ph"( y, z, f, i, m, n, p)    ps"( y, z, f, i, m, n, p)    ch"( y, z, f, i, m, n, p)

Proof of Theorem bnj998
StepHypRef Expression
1 bnj998.4 . . . . . 6  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
2 bnj253 32713 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e.  pred ( y ,  A ,  R ) ) )
32simp1bi 1006 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) )  -> 
( R  FrSe  A  /\  X  e.  A
) )
41, 3sylbi 195 . . . . 5  |-  ( th 
->  ( R  FrSe  A  /\  X  e.  A
) )
54bnj705 32766 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  X  e.  A
) )
6 bnj643 32762 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch )
7 bnj998.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
8 3simpc 990 . . . . . 6  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ( n  =  suc  m  /\  p  =  suc  n ) )
97, 8sylbi 195 . . . . 5  |-  ( ta 
->  ( n  =  suc  m  /\  p  =  suc  n ) )
109bnj707 32768 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( n  =  suc  m  /\  p  =  suc  n ) )
11 bnj255 32714 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ch  /\  n  =  suc  m  /\  p  =  suc  n )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ch  /\  ( n  =  suc  m  /\  p  =  suc  n ) ) )
125, 6, 10, 11syl3anbrc 1175 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ( R  FrSe  A  /\  X  e.  A
)  /\  ch  /\  n  =  suc  m  /\  p  =  suc  n ) )
13 bnj252 32712 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ch  /\  n  =  suc  m  /\  p  =  suc  n )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) )
1412, 13sylib 196 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) )
15 bnj998.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
16 bnj998.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
17 bnj998.3 . . 3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
18 bnj998.7 . . 3  |-  ( ph'  <->  [. p  /  n ]. ph )
19 bnj998.8 . . 3  |-  ( ps'  <->  [. p  /  n ]. ps )
20 bnj998.9 . . 3  |-  ( ch'  <->  [. p  /  n ]. ch )
21 bnj998.10 . . 3  |-  ( ph"  <->  [. G  / 
f ]. ph' )
22 bnj998.11 . . 3  |-  ( ps"  <->  [. G  / 
f ]. ps' )
23 bnj998.12 . . 3  |-  ( ch"  <->  [. G  / 
f ]. ch' )
24 bnj998.13 . . 3  |-  D  =  ( om  \  { (/)
} )
25 bnj998.14 . . 3  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
26 bnj998.15 . . 3  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
27 bnj998.16 . . 3  |-  G  =  ( f  u.  { <. n ,  C >. } )
28 biid 236 . . 3  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
29 biid 236 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
3015, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29bnj910 32962 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ch" )
3114, 30syl 16 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   [.wsbc 3326    \ cdif 3468    u. cun 3469   (/)c0 3780   {csn 4022   <.cop 4028   U_ciun 4320   suc csuc 4875    Fn wfn 5576   ` cfv 5581   omcom 6673    /\ w-bnj17 32695    predc-bnj14 32697    FrSe w-bnj15 32701    trClc-bnj18 32703
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569  ax-reg 8009
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-bnj17 32696  df-bnj14 32698  df-bnj13 32700  df-bnj15 32702
This theorem is referenced by:  bnj1020  32977
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