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Theorem bnj607 29727
Description: Technical lemma for bnj852 29732. 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 572 . . . 4  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( E. m E. p et  /\  th )
)
3 nfv 1761 . . . . . . 7  |-  F/ p th
4319.41 2051 . . . . . 6  |-  ( E. p ( et  /\  th )  <->  ( E. p et  /\  th ) )
54exbii 1718 . . . . 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 29593 . . . . . . 7  |-  ( th 
->  A. m th )
87nfi 1674 . . . . . 6  |-  F/ m th
9819.41 2051 . . . . 5  |-  ( E. m ( E. p et  /\  th )  <->  ( E. m E. p et  /\  th ) )
105, 9bitr2i 254 . . . 4  |-  ( ( E. m E. p et  /\  th )  <->  E. m E. p ( et  /\  th ) )
112, 10sylib 200 . . 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 29615 . . . . . . . . 9  |-  ( et 
->  m  e.  D
)
14 bnj219 29541 . . . . . . . . . 10  |-  ( n  =  suc  m  ->  m  _E  n )
1512, 14bnj770 29574 . . . . . . . . 9  |-  ( et 
->  m  _E  n
)
1613, 15jca 535 . . . . . . . 8  |-  ( et 
->  ( m  e.  D  /\  m  _E  n
) )
1716anim1i 572 . . . . . . 7  |-  ( ( et  /\  th )  ->  ( ( m  e.  D  /\  m  _E  n )  /\  th ) )
18 bnj170 29503 . . . . . . 7  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  <->  ( (
m  e.  D  /\  m  _E  n )  /\  th ) )
1917, 18sylibr 216 . . . . . 6  |-  ( ( et  /\  th )  ->  ( th  /\  m  e.  D  /\  m  _E  n ) )
20 bnj607.38 . . . . . 6  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
2119, 20syl 17 . . . . 5  |-  ( ( et  /\  th )  ->  ch' )
22 simpl 459 . . . . 5  |-  ( ( et  /\  th )  ->  et )
2321, 22jca 535 . . . 4  |-  ( ( et  /\  th )  ->  ( ch'  /\  et ) )
24232eximi 1708 . . 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 198 . . . . . . . . . . 11  |-  ( ch'  ->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
27 euex 2323 . . . . . . . . . . 11  |-  ( E! f ( f  Fn  m  /\  ph'  /\  ps' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
2826, 27syl6 34 . . . . . . . . . 10  |-  ( ch'  ->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
2928impcom 432 . . . . . . . . 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 29607 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ta )
3231adantrr 723 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ch'  /\  et ) )  ->  E. f ta )
33 id 22 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  ta  /\  et ) )
34333com23 1214 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  et  /\  ta )  -> 
( R  FrSe  A  /\  ta  /\  et ) )
35343expia 1210 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  et )  ->  ( ta 
->  ( R  FrSe  A  /\  ta  /\  et ) ) )
3635eximdv 1764 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  et )  ->  ( E. f ta  ->  E. f
( R  FrSe  A  /\  ta  /\  et ) ) )
3736ad2ant2rl 755 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ch'  /\  et ) )  ->  ( E. f ta  ->  E. f
( R  FrSe  A  /\  ta  /\  et ) ) )
3832, 37mpd 15 . . . . . 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 1188 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( G  Fn  n  /\  ph"  /\  ps" ) )
4342eximi 1707 . . . . . 6  |-  ( E. f ( R  FrSe  A  /\  ta  /\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
44 nfe1 1918 . . . . . . 7  |-  F/ f E. f ( f  Fn  n  /\  ph  /\ 
ps )
45 bnj607.28 . . . . . . . . 9  |-  G  e. 
_V
46 nfcv 2592 . . . . . . . . . 10  |-  F/_ h G
47 nfv 1761 . . . . . . . . . . 11  |-  F/ h  G  Fn  n
48 bnj607.300 . . . . . . . . . . . 12  |-  ( ph1  <->  [. G  /  h ]. ph0 )
49 nfsbc1v 3287 . . . . . . . . . . . 12  |-  F/ h [. G  /  h ]. ph0
5048, 49nfxfr 1696 . . . . . . . . . . 11  |-  F/ h ph1
51 bnj607.301 . . . . . . . . . . . 12  |-  ( ps1  <->  [. G  /  h ]. ps0 )
52 nfsbc1v 3287 . . . . . . . . . . . 12  |-  F/ h [. G  /  h ]. ps0
5351, 52nfxfr 1696 . . . . . . . . . . 11  |-  F/ h ps1
5447, 50, 53nf3an 2013 . . . . . . . . . 10  |-  F/ h
( G  Fn  n  /\  ph1  /\  ps1 )
55 fneq1 5664 . . . . . . . . . . 11  |-  ( h  =  G  ->  (
h  Fn  n  <->  G  Fn  n ) )
56 sbceq1a 3278 . . . . . . . . . . . 12  |-  ( h  =  G  ->  ( ph0  <->  [. G  /  h ]. ph0 ) )
5756, 48syl6bbr 267 . . . . . . . . . . 11  |-  ( h  =  G  ->  ( ph0  <->  ph1 ) )
58 sbceq1a 3278 . . . . . . . . . . . 12  |-  ( h  =  G  ->  ( ps0  <->  [. G  /  h ]. ps0 ) )
5958, 51syl6bbr 267 . . . . . . . . . . 11  |-  ( h  =  G  ->  ( ps0  <->  ps1 ) )
6055, 57, 593anbi123d 1339 . . . . . . . . . 10  |-  ( h  =  G  ->  (
( h  Fn  n  /\  ph0  /\  ps0 )  <->  ( G  Fn  n  /\  ph1 
/\  ps1 ) ) )
6146, 54, 60spcegf 3130 . . . . . . . . 9  |-  ( G  e.  _V  ->  (
( G  Fn  n  /\  ph1  /\  ps1 )  ->  E. h ( h  Fn  n  /\  ph0  /\  ps0 )
) )
6245, 61ax-mp 5 . . . . . . . 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 29689 . . . . . . . . . . . . 13  |-  ( ph0  <->  (
h `  (/) )  = 
pred ( x ,  A ,  R ) )
6766, 48, 45bnj526 29699 . . . . . . . . . . . 12  |-  ( ph1  <->  ( G `  (/) )  = 
pred ( x ,  A ,  R ) )
6863, 67bitr4i 256 . . . . . . . . . . 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 3048 . . . . . . . . . . . . . 14  |-  h  e. 
_V
7370, 71, 72bnj540 29703 . . . . . . . . . . . . 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 29703 . . . . . . . . . . . 12  |-  ( ps1  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) ) )
7569, 74bitr4i 256 . . . . . . . . . . 11  |-  ( ps"  <->  ps1 )
7668, 75anbi12i 703 . . . . . . . . . 10  |-  ( ( ph" 
/\  ps" )  <->  ( ph1  /\  ps1 ) )
7776anbi2i 700 . . . . . . . . 9  |-  ( ( G  Fn  n  /\  (
ph"  /\  ps" ) )  <->  ( G  Fn  n  /\  ( ph1 
/\  ps1 ) ) )
78 3anass 989 . . . . . . . . 9  |-  ( ( G  Fn  n  /\  ph"  /\  ps" )  <->  ( G  Fn  n  /\  ( ph"  /\  ps" ) ) )
79 3anass 989 . . . . . . . . 9  |-  ( ( G  Fn  n  /\  ph1 
/\  ps1 )  <->  ( G  Fn  n  /\  ( ph1 
/\  ps1 ) ) )
8077, 78, 793bitr4i 281 . . . . . . . 8  |-  ( ( G  Fn  n  /\  ph"  /\  ps" )  <->  ( G  Fn  n  /\  ph1  /\  ps1 )
)
81 nfv 1761 . . . . . . . . 9  |-  F/ h
( f  Fn  n  /\  ph  /\  ps )
82 nfv 1761 . . . . . . . . . 10  |-  F/ f  h  Fn  n
83 nfsbc1v 3287 . . . . . . . . . . 11  |-  F/ f
[. h  /  f ]. ph
8464, 83nfxfr 1696 . . . . . . . . . 10  |-  F/ f
ph0
85 nfsbc1v 3287 . . . . . . . . . . 11  |-  F/ f
[. h  /  f ]. ps
8671, 85nfxfr 1696 . . . . . . . . . 10  |-  F/ f
ps0
8782, 84, 86nf3an 2013 . . . . . . . . 9  |-  F/ f ( h  Fn  n  /\  ph0  /\  ps0 )
88 fneq1 5664 . . . . . . . . . 10  |-  ( f  =  h  ->  (
f  Fn  n  <->  h  Fn  n ) )
89 sbceq1a 3278 . . . . . . . . . . 11  |-  ( f  =  h  ->  ( ph 
<-> 
[. h  /  f ]. ph ) )
9089, 64syl6bbr 267 . . . . . . . . . 10  |-  ( f  =  h  ->  ( ph 
<-> 
ph0 ) )
91 sbceq1a 3278 . . . . . . . . . . 11  |-  ( f  =  h  ->  ( ps 
<-> 
[. h  /  f ]. ps ) )
9291, 71syl6bbr 267 . . . . . . . . . 10  |-  ( f  =  h  ->  ( ps 
<-> 
ps0 ) )
9388, 90, 923anbi123d 1339 . . . . . . . . 9  |-  ( f  =  h  ->  (
( f  Fn  n  /\  ph  /\  ps )  <->  ( h  Fn  n  /\  ph0 
/\  ps0 ) ) )
9481, 87, 93cbvex 2115 . . . . . . . 8  |-  ( E. f ( f  Fn  n  /\  ph  /\  ps )  <->  E. h ( h  Fn  n  /\  ph0  /\  ps0 )
)
9562, 80, 943imtr4i 270 . . . . . . 7  |-  ( ( G  Fn  n  /\  ph"  /\  ps" )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
9644, 95exlimi 1995 . . . . . 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 437 . . . 4  |-  ( ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
9998exlimivv 1778 . . 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 1203 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 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299    =/= wne 2622   A.wral 2737   _Vcvv 3045   [.wsbc 3267   (/)c0 3731   U_ciun 4278   class class class wbr 4402    _E cep 4743   suc csuc 5425    Fn wfn 5577   ` cfv 5582   omcom 6692   1oc1o 7175    /\ w-bnj17 29491    predc-bnj14 29493    FrSe w-bnj15 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-eprel 4745  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-bnj17 29492
This theorem is referenced by:  bnj600  29730
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