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Theorem bnj554 28601
Description: Technical lemma for bnj852 28623. 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 28512 . 2  |-  ( et 
->  m  =  suc  p )
3 bnj554.20 . . 3  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
43simp3bi 974 . 2  |-  ( ze 
->  m  =  suc  i )
5 simpr 448 . . 3  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  m  =  suc  i )
6 bnj551 28441 . . 3  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  p  =  i )
7 fveq2 5661 . . . 4  |-  ( m  =  suc  i  -> 
( G `  m
)  =  ( G `
 suc  i )
)
8 fveq2 5661 . . . . 5  |-  ( p  =  i  ->  ( G `  p )  =  ( G `  i ) )
9 iuneq1 4041 . . . . . 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 2437 . . . . 5  |-  ( ( G `  p )  =  ( G `  i )  ->  L  =  K )
138, 12syl 16 . . . 4  |-  ( p  =  i  ->  L  =  K )
147, 13eqeqan12d 2395 . . 3  |-  ( ( m  =  suc  i  /\  p  =  i
)  ->  ( ( G `  m )  =  L  <->  ( G `  suc  i )  =  K ) )
155, 6, 14syl2anc 643 . 2  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  (
( G `  m
)  =  L  <->  ( G `  suc  i )  =  K ) )
162, 4, 15syl2an 464 1  |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <-> 
( G `  suc  i )  =  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   U_ciun 4028   suc csuc 4517   omcom 4778   ` cfv 5387    /\ w-bnj17 28381    predc-bnj14 28383
This theorem is referenced by:  bnj558  28604
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634  ax-reg 7486
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-eprel 4428  df-fr 4475  df-suc 4521  df-iota 5351  df-fv 5395  df-bnj17 28382
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