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Theorem bnj605 34347
Description: Technical lemma. 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
bnj605.5  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
bnj605.13  |-  ( ph"  <->  [. f  / 
f ]. ph )
bnj605.14  |-  ( ps"  <->  [. f  / 
f ]. ps )
bnj605.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj605.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj605.28  |-  f  e. 
_V
bnj605.31  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
bnj605.32  |-  ( ph"  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
bnj605.33  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
bnj605.37  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
bnj605.38  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
bnj605.41  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
f  Fn  n )
bnj605.42  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
bnj605.43  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
Assertion
Ref Expression
bnj605  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
Distinct variable groups:    A, f, m    A, p, f    R, f, m    R, p    et, f    m, n    ph, m    ps, m    x, m    n, p    ph, p    ps, p    th, p    x, p
Allowed substitution hints:    ph( x, y, f, i, n)    ps( x, y, f, i, n)    ch( x, y, f, i, m, n, p)    th( x, y, f, i, m, n)    ta( x, y, f, i, m, n, p)    et( x, y, i, m, n, p)    A( x, y, i, n)    D( x, y, f, i, m, n, p)    R( x, y, i, n)    ph'( x, y, f, i, m, n, p)    ps'( x, y, f, i, m, n, p)    ch'( x, y, f, i, m, n, p)    ph"( x, y, f, i, m, n, p)    ps"( x, y, f, i, m, n, p)

Proof of Theorem bnj605
StepHypRef Expression
1 bnj605.37 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
21anim1i 566 . . . 4  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( E. m E. p et  /\  th )
)
3 nfv 1722 . . . . . . 7  |-  F/ p th
4319.41 1989 . . . . . 6  |-  ( E. p ( et  /\  th )  <->  ( E. p et  /\  th ) )
54exbii 1682 . . . . 5  |-  ( E. m E. p ( et  /\  th )  <->  E. m ( E. p et  /\  th ) )
6 bnj605.5 . . . . . . . 8  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
76bnj1095 34222 . . . . . . 7  |-  ( th 
->  A. m th )
87nfi 1638 . . . . . 6  |-  F/ m th
9819.41 1989 . . . . 5  |-  ( E. m ( E. p et  /\  th )  <->  ( E. m E. p et  /\  th ) )
105, 9bitr2i 250 . . . 4  |-  ( ( E. m E. p et  /\  th )  <->  E. m E. p ( et  /\  th ) )
112, 10sylib 196 . . 3  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  ->  E. m E. p ( et  /\  th )
)
12 bnj605.19 . . . . . . . . . 10  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
1312bnj1232 34244 . . . . . . . . 9  |-  ( et 
->  m  e.  D
)
14 bnj219 34170 . . . . . . . . . 10  |-  ( n  =  suc  m  ->  m  _E  n )
1512, 14bnj770 34203 . . . . . . . . 9  |-  ( et 
->  m  _E  n
)
1613, 15jca 530 . . . . . . . 8  |-  ( et 
->  ( m  e.  D  /\  m  _E  n
) )
1716anim1i 566 . . . . . . 7  |-  ( ( et  /\  th )  ->  ( ( m  e.  D  /\  m  _E  n )  /\  th ) )
18 bnj170 34132 . . . . . . 7  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  <->  ( (
m  e.  D  /\  m  _E  n )  /\  th ) )
1917, 18sylibr 212 . . . . . 6  |-  ( ( et  /\  th )  ->  ( th  /\  m  e.  D  /\  m  _E  n ) )
20 bnj605.38 . . . . . 6  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
2119, 20syl 16 . . . . 5  |-  ( ( et  /\  th )  ->  ch' )
22 simpl 455 . . . . 5  |-  ( ( et  /\  th )  ->  et )
2321, 22jca 530 . . . 4  |-  ( ( et  /\  th )  ->  ( ch'  /\  et ) )
24232eximi 1672 . . 3  |-  ( E. m E. p ( et  /\  th )  ->  E. m E. p
( ch'  /\  et ) )
25 bnj248 34134 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  <->  ( (
( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  /\  et ) )
26 bnj605.31 . . . . . . . . . . 11  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
27 pm3.35 585 . . . . . . . . . . 11  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) )
2826, 27sylan2b 473 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) )
29 euex 2254 . . . . . . . . . 10  |-  ( E! f ( f  Fn  m  /\  ph'  /\  ps' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
3028, 29syl 16 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
31 bnj605.17 . . . . . . . . 9  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
3230, 31bnj1198 34236 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ta )
3325, 32bnj832 34197 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ta )
34 bnj605.41 . . . . . . . . . . . . . 14  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
f  Fn  n )
35 bnj605.42 . . . . . . . . . . . . . 14  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
36 bnj605.43 . . . . . . . . . . . . . 14  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
3734, 35, 363jca 1174 . . . . . . . . . . . . 13  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( f  Fn  n  /\  ph"  /\  ps" ) )
38373com23 1200 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  et  /\  ta )  -> 
( f  Fn  n  /\  ph"  /\  ps" ) )
39383expia 1196 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  et )  ->  ( ta 
->  ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4039eximdv 1725 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  et )  ->  ( E. f ta  ->  E. f
( f  Fn  n  /\  ph"  /\  ps" ) ) )
4140adantlr 712 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  et )  ->  ( E. f ta 
->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4241adantlr 712 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  /\  et )  ->  ( E. f ta  ->  E. f
( f  Fn  n  /\  ph"  /\  ps" ) ) )
4325, 42sylbi 195 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  -> 
( E. f ta 
->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4433, 43mpd 15 . . . . . 6  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) )
45 bnj432 34150 . . . . . 6  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  <->  ( ( ch' 
/\  et )  /\  ( R  FrSe  A  /\  x  e.  A )
) )
46 biid 236 . . . . . . . 8  |-  ( f  Fn  n  <->  f  Fn  n )
47 bnj605.13 . . . . . . . . 9  |-  ( ph"  <->  [. f  / 
f ]. ph )
48 sbcid 3282 . . . . . . . . 9  |-  ( [. f  /  f ]. ph  <->  ph )
4947, 48bitri 249 . . . . . . . 8  |-  ( ph"  <->  ph )
50 bnj605.14 . . . . . . . . 9  |-  ( ps"  <->  [. f  / 
f ]. ps )
51 sbcid 3282 . . . . . . . . 9  |-  ( [. f  /  f ]. ps  <->  ps )
5250, 51bitri 249 . . . . . . . 8  |-  ( ps"  <->  ps )
5346, 49, 523anbi123i 1183 . . . . . . 7  |-  ( ( f  Fn  n  /\  ph"  /\  ps" )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
5453exbii 1682 . . . . . 6  |-  ( E. f ( f  Fn  n  /\  ph"  /\  ps" )  <->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
5544, 45, 543imtr3i 265 . . . . 5  |-  ( ( ( ch'  /\  et )  /\  ( R  FrSe  A  /\  x  e.  A
) )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
5655ex 432 . . . 4  |-  ( ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
5756exlimivv 1738 . . 3  |-  ( E. m E. p ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
5811, 24, 573syl 20 . 2  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
59583impa 1189 1  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399   E.wex 1627    e. wcel 1836   E!weu 2228    =/= wne 2587   A.wral 2742   _Vcvv 3047   [.wsbc 3265   (/)c0 3724   U_ciun 4256   class class class wbr 4380    _E cep 4716   suc csuc 4807    Fn wfn 5504   ` cfv 5509   omcom 6617   1oc1o 7059    /\ w-bnj17 34120    predc-bnj14 34122    FrSe w-bnj15 34126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pr 4614
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-br 4381  df-opab 4439  df-eprel 4718  df-suc 4811  df-bnj17 34121
This theorem is referenced by: (None)
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