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Theorem mbfi1fseqlem2 19476
Description: Lemma for mbfi1fseq 19481. (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 9231 . . . . . 6  |-  ( m  =  A  ->  -u m  =  -u A )
2 id 20 . . . . . 6  |-  ( m  =  A  ->  m  =  A )
31, 2oveq12d 6039 . . . . 5  |-  ( m  =  A  ->  ( -u m [,] m )  =  ( -u A [,] A ) )
43eleq2d 2455 . . . 4  |-  ( m  =  A  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u A [,] A
) ) )
5 oveq1 6028 . . . . . 6  |-  ( m  =  A  ->  (
m J x )  =  ( A J x ) )
65, 2breq12d 4167 . . . . 5  |-  ( m  =  A  ->  (
( m J x )  <_  m  <->  ( A J x )  <_  A ) )
76, 5, 2ifbieq12d 3705 . . . 4  |-  ( m  =  A  ->  if ( ( m J x )  <_  m ,  ( m J x ) ,  m
)  =  if ( ( A J x )  <_  A , 
( A J x ) ,  A ) )
8 eqidd 2389 . . . 4  |-  ( m  =  A  ->  0  =  0 )
94, 7, 8ifbieq12d 3705 . . 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 4238 . 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 9015 . . 3  |-  RR  e.  _V
1312mptex 5906 . 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 5746 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 1649    e. wcel 1717   ifcif 3683   class class class wbr 4154    e. cmpt 4208   -->wf 5391   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   RRcr 8923   0cc0 8924    x. cmul 8929    +oocpnf 9051    <_ cle 9055   -ucneg 9225    / cdiv 9610   NNcn 9933   2c2 9982   [,)cico 10851   [,]cicc 10852   |_cfl 11129   ^cexp 11310  MblFncmbf 19374
This theorem is referenced by:  mbfi1fseqlem3  19477  mbfi1fseqlem4  19478  mbfi1fseqlem5  19479  mbfi1fseqlem6  19480
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-cnex 8980  ax-resscn 8981
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 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-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-neg 9227
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