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Theorem mbfi1fseqlem2 19087
Description: Lemma for mbfi1fseq 19092. (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 9060 . . . . . 6  |-  ( m  =  A  ->  -u m  =  -u A )
2 id 19 . . . . . 6  |-  ( m  =  A  ->  m  =  A )
31, 2oveq12d 5892 . . . . 5  |-  ( m  =  A  ->  ( -u m [,] m )  =  ( -u A [,] A ) )
43eleq2d 2363 . . . 4  |-  ( m  =  A  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u A [,] A
) ) )
5 oveq1 5881 . . . . . 6  |-  ( m  =  A  ->  (
m J x )  =  ( A J x ) )
65, 2breq12d 4052 . . . . 5  |-  ( m  =  A  ->  (
( m J x )  <_  m  <->  ( A J x )  <_  A ) )
76, 5, 2ifbieq12d 3600 . . . 4  |-  ( m  =  A  ->  if ( ( m J x )  <_  m ,  ( m J x ) ,  m
)  =  if ( ( A J x )  <_  A , 
( A J x ) ,  A ) )
8 eqidd 2297 . . . 4  |-  ( m  =  A  ->  0  =  0 )
94, 7, 8ifbieq12d 3600 . . 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 4123 . 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 8844 . . 3  |-  RR  e.  _V
1312mptex 5762 . 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 5618 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 1632    e. wcel 1696   ifcif 3578   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   RRcr 8752   0cc0 8753    x. cmul 8758    +oocpnf 8880    <_ cle 8884   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   [,)cico 10674   [,]cicc 10675   |_cfl 10940   ^cexp 11120  MblFncmbf 18985
This theorem is referenced by:  mbfi1fseqlem3  19088  mbfi1fseqlem4  19089  mbfi1fseqlem5  19090  mbfi1fseqlem6  19091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-neg 9056
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