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Theorem bnj1001 29391
Description: Technical lemma for bnj69 29441. 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
bnj1001.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1001.5  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
bnj1001.6  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
bnj1001.13  |-  D  =  ( om  \  { (/)
} )
bnj1001.27  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
Assertion
Ref Expression
bnj1001  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )

Proof of Theorem bnj1001
StepHypRef Expression
1 bnj1001.27 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
2 bnj1001.6 . . . . 5  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
32simplbi 448 . . . 4  |-  ( et 
->  i  e.  n
)
43bnj708 29186 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  n )
5 bnj1001.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
65bnj1232 29237 . . . . 5  |-  ( ch 
->  n  e.  D
)
76bnj706 29184 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  D )
8 bnj1001.13 . . . . 5  |-  D  =  ( om  \  { (/)
} )
98bnj923 29199 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
107, 9syl 16 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  om )
11 elnn 4857 . . 3  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
124, 10, 11syl2anc 644 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  om )
13 bnj1001.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
1413simp3bi 975 . . . . 5  |-  ( ta 
->  p  =  suc  n )
1514bnj707 29185 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  p  =  suc  n )
16 nnord 4855 . . . . . . 7  |-  ( n  e.  om  ->  Ord  n )
17 ordsucelsuc 4804 . . . . . . 7  |-  ( Ord  n  ->  ( i  e.  n  <->  suc  i  e.  suc  n ) )
189, 16, 173syl 19 . . . . . 6  |-  ( n  e.  D  ->  (
i  e.  n  <->  suc  i  e. 
suc  n ) )
1918biimpa 472 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  suc  i  e.  suc  n )
20 eleq2 2499 . . . . 5  |-  ( p  =  suc  n  -> 
( suc  i  e.  p 
<->  suc  i  e.  suc  n ) )
2119, 20anim12i 551 . . . 4  |-  ( ( ( n  e.  D  /\  i  e.  n
)  /\  p  =  suc  n )  ->  ( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e.  suc  n ) ) )
227, 4, 15, 21syl21anc 1184 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e. 
suc  n ) ) )
23 bnj926 29200 . . 3  |-  ( ( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e.  suc  n ) )  ->  suc  i  e.  p
)
2422, 23syl 16 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  p
)
251, 12, 243jca 1135 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319   (/)c0 3630   {csn 3816   Ord word 4582   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29112
This theorem is referenced by:  bnj1020  29396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-bnj17 29113
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