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Theorem bnj600 29267
Description: Technical lemma for bnj852 29269. 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 29237 . . . . . 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 2804 . . . . . . 7  |-  m  e. 
_V
138, 9, 10, 11, 12bnj207 29229 . . . . . 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 29265 . . . . . 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 29266 . . . . . 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 29074 . . . . . . . . 9  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
20 df-rex 2562 . . . . . . . . 9  |-  ( E. m  e.  D  n  =  suc  m  <->  E. m
( m  e.  D  /\  n  =  suc  m ) )
2119, 20sylib 188 . . . . . . . 8  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
2218bnj158 29073 . . . . . . . . . . . . . 14  |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
23 df-rex 2562 . . . . . . . . . . . . . 14  |-  ( E. p  e.  om  m  =  suc  p  <->  E. p
( p  e.  om  /\  m  =  suc  p
) )
2422, 23sylib 188 . . . . . . . . . . . . 13  |-  ( m  e.  D  ->  E. p
( p  e.  om  /\  m  =  suc  p
) )
2524adantr 451 . . . . . . . . . . . 12  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  E. p ( p  e.  om  /\  m  =  suc  p ) )
2625ancri 535 . . . . . . . . . . 11  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  ( E. p
( p  e.  om  /\  m  =  suc  p
)  /\  ( m  e.  D  /\  n  =  suc  m ) ) )
2726bnj534 29084 . . . . . . . . . 10  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  E. p ( ( p  e.  om  /\  m  =  suc  p )  /\  ( m  e.  D  /\  n  =  suc  m ) ) )
28 bnj432 29057 . . . . . . . . . . 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 1572 . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( ( m  e.  D  /\  n  =  suc  m )  ->  E. p ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
3130eximi 1566 . . . . . . . 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 15 . . . . . . 7  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p
( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p
) )
3352exbii 1573 . . . . . . 7  |-  ( E. m E. p et  <->  E. m E. p ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
3432, 33sylibr 203 . . . . . 6  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
35 rsp 2616 . . . . . . . . 9  |-  ( A. m  e.  D  (
m  _E  n  ->  [. m  /  n ]. ch )  ->  (
m  e.  D  -> 
( m  _E  n  ->  [. m  /  n ]. ch ) ) )
361, 35sylbi 187 . . . . . . . 8  |-  ( th 
->  ( m  e.  D  ->  ( m  _E  n  ->  [. m  /  n ]. ch ) ) )
37363imp 1145 . . . . . . 7  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  [. m  /  n ]. ch )
3837, 11sylibr 203 . . . . . 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 29235 . . . . . . . 8  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
4116, 10, 12bnj539 29239 . . . . . . . 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 29242 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
4339, 5, 42bnj561 29251 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
4440, 18, 6, 4, 39, 42, 15bnj545 29243 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
4539, 5, 44bnj562 29252 . . . . . 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 29254 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
54 biid 227 . . . . . 6  |-  ( [. z  /  f ]. ph  <->  [. z  / 
f ]. ph )
55 biid 227 . . . . . 6  |-  ( [. z  /  f ]. ps  <->  [. z  /  f ]. ps )
56 biid 227 . . . . . 6  |-  ( [. G  /  z ]. [. z  /  f ]. ph  <->  [. G  / 
z ]. [. z  / 
f ]. ph )
57 biid 227 . . . . . 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 29264 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
5914, 16, 18bnj579 29262 . . . . . . 7  |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
6059a1d 22 . . . . . 6  |-  ( n  e.  D  ->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  n  /\  ph  /\  ps )
) )
61603ad2ant2 977 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) ) )
6258, 61jcad 519 . . . 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 2194 . . . 4  |-  ( E! f ( f  Fn  n  /\  ph  /\  ps )  <->  ( E. f
( f  Fn  n  /\  ph  /\  ps )  /\  E* f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
6462, 63syl6ibr 218 . . 3  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
6564, 8sylibr 203 . 2  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ch )
66653expib 1154 1  |-  ( n  =/=  1o  ->  (
( n  e.  D  /\  th )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   E*wmo 2157    =/= wne 2459   A.wral 2556   E.wrex 2557   [.wsbc 3004    \ cdif 3162    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656   U_ciun 3921   class class class wbr 4039    _E cep 4319   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271   1oc1o 6488    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj601  29268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034
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