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Theorem bnj600 29290
Description: Technical lemma for bnj852 29292. 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
bnj600.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj600.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj600.3  |-  D  =  ( om  \  { (/)
} )
bnj600.4  |-  ( ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj600.5  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
bnj600.10  |-  ( ph'  <->  [. m  /  n ]. ph )
bnj600.11  |-  ( ps'  <->  [. m  /  n ]. ps )
bnj600.12  |-  ( ch'  <->  [. m  /  n ]. ch )
bnj600.13  |-  ( ph"  <->  [. G  / 
f ]. ph )
bnj600.14  |-  ( ps"  <->  [. G  / 
f ]. ps )
bnj600.15  |-  ( ch"  <->  [. G  / 
f ]. ch )
bnj600.16  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj600.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj600.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj600.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj600.20  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
bnj600.21  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
bnj600.22  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj600.23  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
bnj600.24  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj600.25  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj600.26  |-  G  =  ( f  u.  { <. m ,  C >. } )
Assertion
Ref Expression
bnj600  |-  ( n  =/=  1o  ->  (
( n  e.  D  /\  th )  ->  ch ) )
Distinct variable groups:    A, f,
i, m, n, p   
y, A, f, i, n, p    D, f, p    i, G, y    R, f, i, m, n, p    y, R    et, f, i    x, f, m, n, p    i, ph', p    ph, m, p    ps, m, p    th, 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, m, n, p)    ze( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    rh( x, y, f, i, m, n, p)    A( x)    B( x, y, f, i, m, n, p)    C( x, y, f, i, m, n, p)    D( x, y, i, m, n)    R( x)    G( x, f, m, n, p)    K( x, y, f, i, m, n, p)    L( x, y, f, i, m, n, p)    ph'( x, y, f, m, n)    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)    ch"( x, y, f, i, m, n, p)

Proof of Theorem bnj600
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj600.5 . . . . . 6  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
2 bnj600.13 . . . . . 6  |-  ( ph"  <->  [. G  / 
f ]. ph )
3 bnj600.14 . . . . . 6  |-  ( ps"  <->  [. G  / 
f ]. ps )
4 bnj600.17 . . . . . 6  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
5 bnj600.19 . . . . . 6  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
6 bnj600.16 . . . . . . 7  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
76bnj528 29260 . . . . . 6  |-  G  e. 
_V
8 bnj600.4 . . . . . . 7  |-  ( ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
9 bnj600.10 . . . . . . 7  |-  ( ph'  <->  [. m  /  n ]. ph )
10 bnj600.11 . . . . . . 7  |-  ( ps'  <->  [. m  /  n ]. ps )
11 bnj600.12 . . . . . . 7  |-  ( ch'  <->  [. m  /  n ]. ch )
12 vex 2959 . . . . . . 7  |-  m  e. 
_V
138, 9, 10, 11, 12bnj207 29252 . . . . . 6  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
14 bnj600.1 . . . . . . 7  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
1514, 2, 7bnj609 29288 . . . . . 6  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
16 bnj600.2 . . . . . . 7  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1716, 3, 7bnj611 29289 . . . . . 6  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
18 bnj600.3 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
1918bnj168 29097 . . . . . . . . 9  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
20 df-rex 2711 . . . . . . . . 9  |-  ( E. m  e.  D  n  =  suc  m  <->  E. m
( m  e.  D  /\  n  =  suc  m ) )
2119, 20sylib 189 . . . . . . . 8  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
2218bnj158 29096 . . . . . . . . . . . . . 14  |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
23 df-rex 2711 . . . . . . . . . . . . . 14  |-  ( E. p  e.  om  m  =  suc  p  <->  E. p
( p  e.  om  /\  m  =  suc  p
) )
2422, 23sylib 189 . . . . . . . . . . . . 13  |-  ( m  e.  D  ->  E. p
( p  e.  om  /\  m  =  suc  p
) )
2524adantr 452 . . . . . . . . . . . 12  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  E. p ( p  e.  om  /\  m  =  suc  p ) )
2625ancri 536 . . . . . . . . . . 11  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  ( E. p
( p  e.  om  /\  m  =  suc  p
)  /\  ( m  e.  D  /\  n  =  suc  m ) ) )
2726bnj534 29107 . . . . . . . . . 10  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  E. p ( ( p  e.  om  /\  m  =  suc  p )  /\  ( m  e.  D  /\  n  =  suc  m ) ) )
28 bnj432 29080 . . . . . . . . . . 11  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  ( (
p  e.  om  /\  m  =  suc  p )  /\  ( m  e.  D  /\  n  =  suc  m ) ) )
2928exbii 1592 . . . . . . . . . 10  |-  ( E. p ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  E. p
( ( p  e. 
om  /\  m  =  suc  p )  /\  (
m  e.  D  /\  n  =  suc  m ) ) )
3027, 29sylibr 204 . . . . . . . . 9  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  E. p ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
3130eximi 1585 . . . . . . . 8  |-  ( E. m ( m  e.  D  /\  n  =  suc  m )  ->  E. m E. p ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
3221, 31syl 16 . . . . . . 7  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p
( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p
) )
3352exbii 1593 . . . . . . 7  |-  ( E. m E. p et  <->  E. m E. p ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
3432, 33sylibr 204 . . . . . 6  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
35 rsp 2766 . . . . . . . . 9  |-  ( A. m  e.  D  (
m  _E  n  ->  [. m  /  n ]. ch )  ->  (
m  e.  D  -> 
( m  _E  n  ->  [. m  /  n ]. ch ) ) )
361, 35sylbi 188 . . . . . . . 8  |-  ( th 
->  ( m  e.  D  ->  ( m  _E  n  ->  [. m  /  n ]. ch ) ) )
37363imp 1147 . . . . . . 7  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  [. m  /  n ]. ch )
3837, 11sylibr 204 . . . . . 6  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
39 bnj600.18 . . . . . . 7  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
4014, 9, 12bnj523 29258 . . . . . . . 8  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
4116, 10, 12bnj539 29262 . . . . . . . 8  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
4240, 41, 18, 6, 4, 39bnj544 29265 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
4339, 5, 42bnj561 29274 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
4440, 18, 6, 4, 39, 42, 15bnj545 29266 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
4539, 5, 44bnj562 29275 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
46 bnj600.20 . . . . . . 7  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
47 bnj600.22 . . . . . . 7  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
48 bnj600.23 . . . . . . 7  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
49 bnj600.24 . . . . . . 7  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
50 bnj600.25 . . . . . . 7  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
51 bnj600.26 . . . . . . 7  |-  G  =  ( f  u.  { <. m ,  C >. } )
52 bnj600.21 . . . . . . 7  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
5318, 6, 4, 39, 5, 46, 47, 48, 49, 50, 51, 40, 41, 42, 52, 43, 17bnj571 29277 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
54 biid 228 . . . . . 6  |-  ( [. z  /  f ]. ph  <->  [. z  / 
f ]. ph )
55 biid 228 . . . . . 6  |-  ( [. z  /  f ]. ps  <->  [. z  /  f ]. ps )
56 biid 228 . . . . . 6  |-  ( [. G  /  z ]. [. z  /  f ]. ph  <->  [. G  / 
z ]. [. z  / 
f ]. ph )
57 biid 228 . . . . . 6  |-  ( [. G  /  z ]. [. z  /  f ]. ps  <->  [. G  /  z ]. [. z  /  f ]. ps )
581, 2, 3, 4, 5, 7, 13, 15, 17, 34, 38, 43, 45, 53, 14, 16, 54, 55, 56, 57bnj607 29287 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
5914, 16, 18bnj579 29285 . . . . . . 7  |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
6059a1d 23 . . . . . 6  |-  ( n  e.  D  ->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  n  /\  ph  /\  ps )
) )
61603ad2ant2 979 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) ) )
6258, 61jcad 520 . . . 4  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( E. f ( f  Fn  n  /\  ph  /\  ps )  /\  E* f
( f  Fn  n  /\  ph  /\  ps )
) ) )
63 eu5 2319 . . . 4  |-  ( E! f ( f  Fn  n  /\  ph  /\  ps )  <->  ( E. f
( f  Fn  n  /\  ph  /\  ps )  /\  E* f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
6462, 63syl6ibr 219 . . 3  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
6564, 8sylibr 204 . 2  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ch )
66653expib 1156 1  |-  ( n  =/=  1o  ->  (
( n  e.  D  /\  th )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   E*wmo 2282    =/= wne 2599   A.wral 2705   E.wrex 2706   [.wsbc 3161    \ cdif 3317    u. cun 3318   (/)c0 3628   {csn 3814   <.cop 3817   U_ciun 4093   class class class wbr 4212    _E cep 4492   suc csuc 4583   omcom 4845    Fn wfn 5449   ` cfv 5454   1oc1o 6717    /\ w-bnj17 29050    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj601  29291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057
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