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Theorem bnj998 28988
Description: Technical lemma for bnj69 29040. 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 28729 . . . . . . 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 970 . . . . . 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 187 . . . . 5  |-  ( th 
->  ( R  FrSe  A  /\  X  e.  A
) )
54bnj705 28782 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  X  e.  A
) )
6 bnj643 28778 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch )
7 bnj998.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
8 3simpc 954 . . . . . 6  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ( n  =  suc  m  /\  p  =  suc  n ) )
97, 8sylbi 187 . . . . 5  |-  ( ta 
->  ( n  =  suc  m  /\  p  =  suc  n ) )
109bnj707 28784 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( n  =  suc  m  /\  p  =  suc  n ) )
11 bnj255 28730 . . . 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 1136 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ( R  FrSe  A  /\  X  e.  A
)  /\  ch  /\  n  =  suc  m  /\  p  =  suc  n ) )
13 bnj252 28728 . . 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 188 . 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 227 . . 3  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
29 biid 227 . . 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 28980 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ch" )
3114, 30syl 15 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   [.wsbc 2991    \ cdif 3149    u. cun 3150   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1020  28995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718
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