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Theorem bnj605 33833
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 568 . . . 4  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( E. m E. p et  /\  th )
)
3 nfv 1694 . . . . . . 7  |-  F/ p th
4319.41 1957 . . . . . 6  |-  ( E. p ( et  /\  th )  <->  ( E. p et  /\  th ) )
54exbii 1654 . . . . 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 33708 . . . . . . 7  |-  ( th 
->  A. m th )
87nfi 1610 . . . . . 6  |-  F/ m th
9819.41 1957 . . . . 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 33730 . . . . . . . . 9  |-  ( et 
->  m  e.  D
)
14 bnj219 33656 . . . . . . . . . 10  |-  ( n  =  suc  m  ->  m  _E  n )
1512, 14bnj770 33689 . . . . . . . . 9  |-  ( et 
->  m  _E  n
)
1613, 15jca 532 . . . . . . . 8  |-  ( et 
->  ( m  e.  D  /\  m  _E  n
) )
1716anim1i 568 . . . . . . 7  |-  ( ( et  /\  th )  ->  ( ( m  e.  D  /\  m  _E  n )  /\  th ) )
18 bnj170 33618 . . . . . . 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 457 . . . . 5  |-  ( ( et  /\  th )  ->  et )
2321, 22jca 532 . . . 4  |-  ( ( et  /\  th )  ->  ( ch'  /\  et ) )
24232eximi 1644 . . 3  |-  ( E. m E. p ( et  /\  th )  ->  E. m E. p
( ch'  /\  et ) )
25 bnj248 33620 . . . . . . . 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 pm3.35 587 . . . . . . . . . . 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 475 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) )
29 euex 2294 . . . . . . . . . 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 33722 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ta )
3325, 32bnj832 33683 . . . . . . 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 1177 . . . . . . . . . . . . 13  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( f  Fn  n  /\  ph"  /\  ps" ) )
38373com23 1203 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  et  /\  ta )  -> 
( f  Fn  n  /\  ph"  /\  ps" ) )
39383expia 1199 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  et )  ->  ( ta 
->  ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4039eximdv 1697 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  et )  ->  ( E. f ta  ->  E. f
( f  Fn  n  /\  ph"  /\  ps" ) ) )
4140adantlr 714 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  et )  ->  ( E. f ta 
->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4241adantlr 714 . . . . . . . 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 33636 . . . . . 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 3330 . . . . . . . . 9  |-  ( [. f  /  f ]. ph  <->  ph )
4947, 48bitri 249 . . . . . . . 8  |-  ( ph"  <->  ph )
50 bnj605.14 . . . . . . . . 9  |-  ( ps"  <->  [. f  / 
f ]. ps )
51 sbcid 3330 . . . . . . . . 9  |-  ( [. f  /  f ]. ps  <->  ps )
5250, 51bitri 249 . . . . . . . 8  |-  ( ps"  <->  ps )
5346, 49, 523anbi123i 1186 . . . . . . 7  |-  ( ( f  Fn  n  /\  ph"  /\  ps" )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
5453exbii 1654 . . . . . 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 434 . . . 4  |-  ( ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
5756exlimivv 1710 . . 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 1192 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 369    /\ w3a 974    = wceq 1383   E.wex 1599    e. wcel 1804   E!weu 2268    =/= wne 2638   A.wral 2793   _Vcvv 3095   [.wsbc 3313   (/)c0 3770   U_ciun 4315   class class class wbr 4437    _E cep 4779   suc csuc 4870    Fn wfn 5573   ` cfv 5578   omcom 6685   1oc1o 7125    /\ w-bnj17 33606    predc-bnj14 33608    FrSe w-bnj15 33612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-eprel 4781  df-suc 4874  df-bnj17 33607
This theorem is referenced by: (None)
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