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Theorem bnj554 28931
Description: Technical lemma for bnj852 28953. 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
bnj554.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj554.20  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
bnj554.21  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj554.22  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj554.23  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj554.24  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
Assertion
Ref Expression
bnj554  |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <-> 
( G `  suc  i )  =  K ) )
Distinct variable groups:    y, G    y, i    y, p
Allowed substitution hints:    et( y, i, m, n, p)    ze( y,
i, m, n, p)    A( y, i, m, n, p)    D( y, i, m, n, p)    R( y,
i, m, n, p)    G( i, m, n, p)    K( y, i, m, n, p)    L( y, i, m, n, p)

Proof of Theorem bnj554
StepHypRef Expression
1 bnj554.19 . . 3  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
21bnj1254 28842 . 2  |-  ( et 
->  m  =  suc  p )
3 bnj554.20 . . 3  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
43simp3bi 972 . 2  |-  ( ze 
->  m  =  suc  i )
5 simpr 447 . . 3  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  m  =  suc  i )
6 bnj551 28771 . . 3  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  p  =  i )
7 fveq2 5525 . . . 4  |-  ( m  =  suc  i  -> 
( G `  m
)  =  ( G `
 suc  i )
)
8 fveq2 5525 . . . . 5  |-  ( p  =  i  ->  ( G `  p )  =  ( G `  i ) )
9 iuneq1 3918 . . . . . 6  |-  ( ( G `  p )  =  ( G `  i )  ->  U_ y  e.  ( G `  p
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
10 bnj554.24 . . . . . 6  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
11 bnj554.23 . . . . . 6  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
129, 10, 113eqtr4g 2340 . . . . 5  |-  ( ( G `  p )  =  ( G `  i )  ->  L  =  K )
138, 12syl 15 . . . 4  |-  ( p  =  i  ->  L  =  K )
147, 13eqeqan12d 2298 . . 3  |-  ( ( m  =  suc  i  /\  p  =  i
)  ->  ( ( G `  m )  =  L  <->  ( G `  suc  i )  =  K ) )
155, 6, 14syl2anc 642 . 2  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  (
( G `  m
)  =  L  <->  ( G `  suc  i )  =  K ) )
162, 4, 15syl2an 463 1  |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <-> 
( G `  suc  i )  =  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   U_ciun 3905   suc csuc 4394   omcom 4656   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713
This theorem is referenced by:  bnj558  28934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352  df-suc 4398  df-iota 5219  df-fv 5263  df-bnj17 28712
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