Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj998 Structured version   Unicode version

Theorem bnj998 29329
Description: Technical lemma for bnj69 29381. 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 29070 . . . . . . 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 973 . . . . . 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 189 . . . . 5  |-  ( th 
->  ( R  FrSe  A  /\  X  e.  A
) )
54bnj705 29123 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  X  e.  A
) )
6 bnj643 29119 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch )
7 bnj998.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
8 3simpc 957 . . . . . 6  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ( n  =  suc  m  /\  p  =  suc  n ) )
97, 8sylbi 189 . . . . 5  |-  ( ta 
->  ( n  =  suc  m  /\  p  =  suc  n ) )
109bnj707 29125 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( n  =  suc  m  /\  p  =  suc  n ) )
11 bnj255 29071 . . . 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 1139 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ( R  FrSe  A  /\  X  e.  A
)  /\  ch  /\  n  =  suc  m  /\  p  =  suc  n ) )
13 bnj252 29069 . . 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 190 . 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 229 . . 3  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
29 biid 229 . . 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 29321 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   [.wsbc 3163    \ cdif 3319    u. cun 3320   (/)c0 3630   {csn 3816   <.cop 3819   U_ciun 4095   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29052    predc-bnj14 29054    FrSe w-bnj15 29058    trClc-bnj18 29060
This theorem is referenced by:  bnj1020  29336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-bnj17 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059
  Copyright terms: Public domain W3C validator