Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1001 Unicode version

Theorem bnj1001 28668
Description: Technical lemma for bnj69 28718. 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 447 . . . 4  |-  ( et 
->  i  e.  n
)
43bnj708 28463 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  n )
5 bnj1001.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
65bnj1232 28514 . . . . 5  |-  ( ch 
->  n  e.  D
)
76bnj706 28461 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  D )
8 bnj1001.13 . . . . 5  |-  D  =  ( om  \  { (/)
} )
98bnj923 28476 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
107, 9syl 16 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  om )
11 elnn 4796 . . 3  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
124, 10, 11syl2anc 643 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  om )
13 bnj1001.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
1413simp3bi 974 . . . . 5  |-  ( ta 
->  p  =  suc  n )
1514bnj707 28462 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  p  =  suc  n )
16 nnord 4794 . . . . . . 7  |-  ( n  e.  om  ->  Ord  n )
17 ordsucelsuc 4743 . . . . . . 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 471 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  suc  i  e.  suc  n )
20 eleq2 2449 . . . . 5  |-  ( p  =  suc  n  -> 
( suc  i  e.  p 
<->  suc  i  e.  suc  n ) )
2119, 20anim12i 550 . . . 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 1183 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e. 
suc  n ) ) )
23 bnj926 28477 . . 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 1134 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    \ cdif 3261   (/)c0 3572   {csn 3758   Ord word 4522   suc csuc 4525   omcom 4786    Fn wfn 5390   ` cfv 5395    /\ w-bnj17 28389
This theorem is referenced by:  bnj1020  28673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-bnj17 28390
  Copyright terms: Public domain W3C validator