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Theorem mbfi1fseqlem2 19600
Description: Lemma for mbfi1fseq 19605. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1  |-  ( ph  ->  F  e. MblFn )
mbfi1fseq.2  |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )
mbfi1fseq.3  |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) ) )
mbfi1fseq.4  |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m J x )  <_  m , 
( m J x ) ,  m ) ,  0 ) ) )
Assertion
Ref Expression
mbfi1fseqlem2  |-  ( A  e.  NN  ->  ( G `  A )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A
) ,  0 ) ) )
Distinct variable groups:    x, m, y, F    x, G    m, J    ph, m, x, y    A, m, x, y
Allowed substitution hints:    G( y, m)    J( x, y)

Proof of Theorem mbfi1fseqlem2
StepHypRef Expression
1 negeq 9290 . . . . . 6  |-  ( m  =  A  ->  -u m  =  -u A )
2 id 20 . . . . . 6  |-  ( m  =  A  ->  m  =  A )
31, 2oveq12d 6091 . . . . 5  |-  ( m  =  A  ->  ( -u m [,] m )  =  ( -u A [,] A ) )
43eleq2d 2502 . . . 4  |-  ( m  =  A  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u A [,] A
) ) )
5 oveq1 6080 . . . . . 6  |-  ( m  =  A  ->  (
m J x )  =  ( A J x ) )
65, 2breq12d 4217 . . . . 5  |-  ( m  =  A  ->  (
( m J x )  <_  m  <->  ( A J x )  <_  A ) )
76, 5, 2ifbieq12d 3753 . . . 4  |-  ( m  =  A  ->  if ( ( m J x )  <_  m ,  ( m J x ) ,  m
)  =  if ( ( A J x )  <_  A , 
( A J x ) ,  A ) )
8 eqidd 2436 . . . 4  |-  ( m  =  A  ->  0  =  0 )
94, 7, 8ifbieq12d 3753 . . 3  |-  ( m  =  A  ->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x )  <_  m ,  ( m J x ) ,  m ) ,  0 )  =  if ( x  e.  (
-u A [,] A
) ,  if ( ( A J x )  <_  A , 
( A J x ) ,  A ) ,  0 ) )
109mpteq2dv 4288 . 2  |-  ( m  =  A  ->  (
x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m J x )  <_  m , 
( m J x ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A
) ,  0 ) ) )
11 mbfi1fseq.4 . 2  |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m J x )  <_  m , 
( m J x ) ,  m ) ,  0 ) ) )
12 reex 9073 . . 3  |-  RR  e.  _V
1312mptex 5958 . 2  |-  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A ) ,  0 ) )  e.  _V
1410, 11, 13fvmpt 5798 1  |-  ( A  e.  NN  ->  ( G `  A )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A
) ,  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   RRcr 8981   0cc0 8982    x. cmul 8987    +oocpnf 9109    <_ cle 9113   -ucneg 9284    / cdiv 9669   NNcn 9992   2c2 10041   [,)cico 10910   [,]cicc 10911   |_cfl 11193   ^cexp 11374  MblFncmbf 19498
This theorem is referenced by:  mbfi1fseqlem3  19601  mbfi1fseqlem4  19602  mbfi1fseqlem5  19603  mbfi1fseqlem6  19604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-neg 9286
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