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Theorem bnj607 28948
Description: Technical lemma for bnj852 28953. 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
bnj607.5  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
bnj607.13  |-  ( ph"  <->  [. G  / 
f ]. ph )
bnj607.14  |-  ( ps"  <->  [. G  / 
f ]. ps )
bnj607.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj607.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj607.28  |-  G  e. 
_V
bnj607.31  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
bnj607.32  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
bnj607.33  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
bnj607.37  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
bnj607.38  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
bnj607.41  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
bnj607.42  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
bnj607.43  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
bnj607.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj607.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj607.400  |-  ( ph0  <->  [. h  /  f ]. ph )
bnj607.401  |-  ( ps0  <->  [. h  /  f ]. ps )
bnj607.300  |-  ( ph1  <->  [. G  /  h ]. ph0 )
bnj607.301  |-  ( ps1  <->  [. G  /  h ]. ps0 )
Assertion
Ref Expression
bnj607  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
Distinct variable groups:    A, f, h    A, m, f    A, p, f    h, G, i, y    R, f, h    R, m    R, p    et, f    f, i, y    f, n, h    x, f, h    ph, h    ps, h    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, h, i, m, n, p)    th( x, y, f, h, i, m, n)    ta( x, y, f, h, i, m, n, p)    et( x, y, h, i, m, n, p)    A( x, y, i, n)    D( x, y, f, h, i, m, n, p)    R( x, y, i, n)    G( x, f, m, n, p)    ph'( x, y, f, h, i, m, n, p)    ps'( x, y, f, h, i, m, n, p)    ch'( x, y, f, h, i, m, n, p)    ph"( x, y, f, h, i, m, n, p)   
ps"( x, y, f, h, i, m, n, p)    ph0( x, y, f, h, i, m, n, p)    ps0( x, y, f, h, i, m, n, p)    ph1( x, y, f, h, i, m, n, p)    ps1( x, y, f, h, i, m, n, p)

Proof of Theorem bnj607
StepHypRef Expression
1 bnj607.37 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
21anim1i 551 . . . 4  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( E. m E. p et  /\  th )
)
3 nfv 1605 . . . . . . 7  |-  F/ p th
4319.41 1815 . . . . . 6  |-  ( E. p ( et  /\  th )  <->  ( E. p et  /\  th ) )
54exbii 1569 . . . . 5  |-  ( E. m E. p ( et  /\  th )  <->  E. m ( E. p et  /\  th ) )
6 bnj607.5 . . . . . . . 8  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
76bnj1095 28813 . . . . . . 7  |-  ( th 
->  A. m th )
87nfi 1538 . . . . . 6  |-  F/ m th
9819.41 1815 . . . . 5  |-  ( E. m ( E. p et  /\  th )  <->  ( E. m E. p et  /\  th ) )
105, 9bitr2i 241 . . . 4  |-  ( ( E. m E. p et  /\  th )  <->  E. m E. p ( et  /\  th ) )
112, 10sylib 188 . . 3  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  ->  E. m E. p ( et  /\  th )
)
12 bnj607.19 . . . . . . . . . 10  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
1312bnj1232 28836 . . . . . . . . 9  |-  ( et 
->  m  e.  D
)
14 bnj219 28761 . . . . . . . . . 10  |-  ( n  =  suc  m  ->  m  _E  n )
1512, 14bnj770 28793 . . . . . . . . 9  |-  ( et 
->  m  _E  n
)
1613, 15jca 518 . . . . . . . 8  |-  ( et 
->  ( m  e.  D  /\  m  _E  n
) )
1716anim1i 551 . . . . . . 7  |-  ( ( et  /\  th )  ->  ( ( m  e.  D  /\  m  _E  n )  /\  th ) )
18 bnj170 28723 . . . . . . 7  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  <->  ( (
m  e.  D  /\  m  _E  n )  /\  th ) )
1917, 18sylibr 203 . . . . . 6  |-  ( ( et  /\  th )  ->  ( th  /\  m  e.  D  /\  m  _E  n ) )
20 bnj607.38 . . . . . 6  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
2119, 20syl 15 . . . . 5  |-  ( ( et  /\  th )  ->  ch' )
22 simpl 443 . . . . 5  |-  ( ( et  /\  th )  ->  et )
2321, 22jca 518 . . . 4  |-  ( ( et  /\  th )  ->  ( ch'  /\  et ) )
24232eximi 1564 . . 3  |-  ( E. m E. p ( et  /\  th )  ->  E. m E. p
( ch'  /\  et ) )
25 bnj607.31 . . . . . . . . . . . 12  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
2625biimpi 186 . . . . . . . . . . 11  |-  ( ch'  ->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
27 euex 2166 . . . . . . . . . . 11  |-  ( E! f ( f  Fn  m  /\  ph'  /\  ps' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
2826, 27syl6 29 . . . . . . . . . 10  |-  ( ch'  ->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
2928impcom 419 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
30 bnj607.17 . . . . . . . . 9  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
3129, 30bnj1198 28828 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ta )
3231adantrr 697 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ch'  /\  et ) )  ->  E. f ta )
33 id 19 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  ta  /\  et ) )
34333com23 1157 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  et  /\  ta )  -> 
( R  FrSe  A  /\  ta  /\  et ) )
35343expia 1153 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  et )  ->  ( ta 
->  ( R  FrSe  A  /\  ta  /\  et ) ) )
3635eximdv 1608 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  et )  ->  ( E. f ta  ->  E. f
( R  FrSe  A  /\  ta  /\  et ) ) )
3736ad2ant2rl 729 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ch'  /\  et ) )  ->  ( E. f ta  ->  E. f
( R  FrSe  A  /\  ta  /\  et ) ) )
3832, 37mpd 14 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ch'  /\  et ) )  ->  E. f
( R  FrSe  A  /\  ta  /\  et ) )
39 bnj607.41 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
40 bnj607.42 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
41 bnj607.43 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
4239, 40, 413jca 1132 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( G  Fn  n  /\  ph"  /\  ps" ) )
4342eximi 1563 . . . . . 6  |-  ( E. f ( R  FrSe  A  /\  ta  /\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
44 nfe1 1706 . . . . . . 7  |-  F/ f E. f ( f  Fn  n  /\  ph  /\ 
ps )
45 bnj607.28 . . . . . . . . 9  |-  G  e. 
_V
46 nfcv 2419 . . . . . . . . . 10  |-  F/_ h G
47 nfv 1605 . . . . . . . . . . 11  |-  F/ h  G  Fn  n
48 bnj607.300 . . . . . . . . . . . 12  |-  ( ph1  <->  [. G  /  h ]. ph0 )
49 nfsbc1v 3010 . . . . . . . . . . . 12  |-  F/ h [. G  /  h ]. ph0
5048, 49nfxfr 1557 . . . . . . . . . . 11  |-  F/ h ph1
51 bnj607.301 . . . . . . . . . . . 12  |-  ( ps1  <->  [. G  /  h ]. ps0 )
52 nfsbc1v 3010 . . . . . . . . . . . 12  |-  F/ h [. G  /  h ]. ps0
5351, 52nfxfr 1557 . . . . . . . . . . 11  |-  F/ h ps1
5447, 50, 53nf3an 1774 . . . . . . . . . 10  |-  F/ h
( G  Fn  n  /\  ph1  /\  ps1 )
55 fneq1 5333 . . . . . . . . . . 11  |-  ( h  =  G  ->  (
h  Fn  n  <->  G  Fn  n ) )
56 sbceq1a 3001 . . . . . . . . . . . 12  |-  ( h  =  G  ->  ( ph0  <->  [. G  /  h ]. ph0 ) )
5756, 48syl6bbr 254 . . . . . . . . . . 11  |-  ( h  =  G  ->  ( ph0  <->  ph1 ) )
58 sbceq1a 3001 . . . . . . . . . . . 12  |-  ( h  =  G  ->  ( ps0  <->  [. G  /  h ]. ps0 ) )
5958, 51syl6bbr 254 . . . . . . . . . . 11  |-  ( h  =  G  ->  ( ps0  <->  ps1 ) )
6055, 57, 593anbi123d 1252 . . . . . . . . . 10  |-  ( h  =  G  ->  (
( h  Fn  n  /\  ph0  /\  ps0 )  <->  ( G  Fn  n  /\  ph1 
/\  ps1 ) ) )
6146, 54, 60spcegf 2864 . . . . . . . . 9  |-  ( G  e.  _V  ->  (
( G  Fn  n  /\  ph1  /\  ps1 )  ->  E. h ( h  Fn  n  /\  ph0  /\  ps0 )
) )
6245, 61ax-mp 8 . . . . . . . 8  |-  ( ( G  Fn  n  /\  ph1 
/\  ps1 )  ->  E. h
( h  Fn  n  /\  ph0  /\  ps0 )
)
63 bnj607.32 . . . . . . . . . . . 12  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
64 bnj607.400 . . . . . . . . . . . . . 14  |-  ( ph0  <->  [. h  /  f ]. ph )
65 bnj607.1 . . . . . . . . . . . . . 14  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
6664, 65bnj154 28910 . . . . . . . . . . . . 13  |-  ( ph0  <->  (
h `  (/) )  = 
pred ( x ,  A ,  R ) )
6766, 48, 45bnj526 28920 . . . . . . . . . . . 12  |-  ( ph1  <->  ( G `  (/) )  = 
pred ( x ,  A ,  R ) )
6863, 67bitr4i 243 . . . . . . . . . . 11  |-  ( ph"  <->  ph1 )
69 bnj607.33 . . . . . . . . . . . 12  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
70 bnj607.2 . . . . . . . . . . . . . 14  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
71 bnj607.401 . . . . . . . . . . . . . 14  |-  ( ps0  <->  [. h  /  f ]. ps )
72 vex 2791 . . . . . . . . . . . . . 14  |-  h  e. 
_V
7370, 71, 72bnj540 28924 . . . . . . . . . . . . 13  |-  ( ps0  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( h `  suc  i )  =  U_ y  e.  ( h `  i )  pred (
y ,  A ,  R ) ) )
7473, 51, 45bnj540 28924 . . . . . . . . . . . 12  |-  ( ps1  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) ) )
7569, 74bitr4i 243 . . . . . . . . . . 11  |-  ( ps"  <->  ps1 )
7668, 75anbi12i 678 . . . . . . . . . 10  |-  ( ( ph" 
/\  ps" )  <->  ( ph1  /\  ps1 ) )
7776anbi2i 675 . . . . . . . . 9  |-  ( ( G  Fn  n  /\  (
ph"  /\  ps" ) )  <->  ( G  Fn  n  /\  ( ph1 
/\  ps1 ) ) )
78 3anass 938 . . . . . . . . 9  |-  ( ( G  Fn  n  /\  ph"  /\  ps" )  <->  ( G  Fn  n  /\  ( ph"  /\  ps" ) ) )
79 3anass 938 . . . . . . . . 9  |-  ( ( G  Fn  n  /\  ph1 
/\  ps1 )  <->  ( G  Fn  n  /\  ( ph1 
/\  ps1 ) ) )
8077, 78, 793bitr4i 268 . . . . . . . 8  |-  ( ( G  Fn  n  /\  ph"  /\  ps" )  <->  ( G  Fn  n  /\  ph1  /\  ps1 )
)
81 nfv 1605 . . . . . . . . 9  |-  F/ h
( f  Fn  n  /\  ph  /\  ps )
82 nfv 1605 . . . . . . . . . 10  |-  F/ f  h  Fn  n
83 nfsbc1v 3010 . . . . . . . . . . 11  |-  F/ f
[. h  /  f ]. ph
8464, 83nfxfr 1557 . . . . . . . . . 10  |-  F/ f
ph0
85 nfsbc1v 3010 . . . . . . . . . . 11  |-  F/ f
[. h  /  f ]. ps
8671, 85nfxfr 1557 . . . . . . . . . 10  |-  F/ f
ps0
8782, 84, 86nf3an 1774 . . . . . . . . 9  |-  F/ f ( h  Fn  n  /\  ph0  /\  ps0 )
88 fneq1 5333 . . . . . . . . . 10  |-  ( f  =  h  ->  (
f  Fn  n  <->  h  Fn  n ) )
89 sbceq1a 3001 . . . . . . . . . . 11  |-  ( f  =  h  ->  ( ph 
<-> 
[. h  /  f ]. ph ) )
9089, 64syl6bbr 254 . . . . . . . . . 10  |-  ( f  =  h  ->  ( ph 
<-> 
ph0 ) )
91 sbceq1a 3001 . . . . . . . . . . 11  |-  ( f  =  h  ->  ( ps 
<-> 
[. h  /  f ]. ps ) )
9291, 71syl6bbr 254 . . . . . . . . . 10  |-  ( f  =  h  ->  ( ps 
<-> 
ps0 ) )
9388, 90, 923anbi123d 1252 . . . . . . . . 9  |-  ( f  =  h  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( h  Fn  n  /\  ph0 
/\  ps0 ) ) )
9481, 87, 93cbvex 1925 . . . . . . . 8  |-  ( E. f ( f  Fn  n  /\  ph  /\  ps )  <->  E. h ( h  Fn  n  /\  ph0  /\  ps0 )
)
9562, 80, 943imtr4i 257 . . . . . . 7  |-  ( ( G  Fn  n  /\  ph"  /\  ps" )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
9644, 95exlimi 1801 . . . . . 6  |-  ( E. f ( G  Fn  n  /\  ph"  /\  ps" )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) )
9738, 43, 963syl 18 . . . . 5  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ch'  /\  et ) )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
9897expcom 424 . . . 4  |-  ( ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
9998exlimivv 1667 . . 3  |-  ( E. m E. p ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
10011, 24, 993syl 18 . 2  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
1011003impa 1146 1  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143    =/= wne 2446   A.wral 2543   _Vcvv 2788   [.wsbc 2991   (/)c0 3455   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255   1oc1o 6472    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj600  28951
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-eprel 4305  df-suc 4398  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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