Theorem mbfi1fseqlem2 23483

Description: Lemma for mbfi1fseq 23488. (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

Step	Hyp	Ref	Expression
1		negeq 10273	. . . . . 6 |- ( m = A -> -u m = -u A )
2		id 22	. . . . . 6 |- ( m = A -> m = A )
3	1, 2	oveq12d 6668	. . . . 5 |- ( m = A -> ( -u m [,] m ) = ( -u A [,] A ) )
4	3	eleq2d 2687	. . . 4 |- ( m = A -> ( x e. ( -u m [,] m ) <-> x e. ( -u A [,] A ) ) )
5		oveq1 6657	. . . . . 6 |- ( m = A -> ( m J x ) = ( A J x ) )
6	5, 2	breq12d 4666	. . . . 5 |- ( m = A -> ( ( m J x ) <_ m <-> ( A J x ) <_ A ) )
7	6, 5, 2	ifbieq12d 4113	. . . 4 |- ( m = A -> if ( ( m J x ) <_ m , ( m J x ) , m ) = if ( ( A J x ) <_ A , ( A J x ) , A ) )
8	4, 7	ifbieq1d 4109	. . 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 ) )
9	8	mpteq2dv 4745	. 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 ) ) )
10		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 ) ) )
11		reex 10027	. . 3 |- RR e. _V
12	11	mptex 6486	. 2 |- ( x e. RR |-> if ( x e. ( -u A [,] A ) , if ( ( A J x ) <_ A , ( A J x ) , A ) , 0 ) ) e. _V
13	9, 10, 12	fvmpt 6282	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 ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   <_ cle 10075   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   [,)cico 12177   [,]cicc 12178   |_cfl 12591   ^cexp 12860  MblFncmbf 23383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-neg 10269

This theorem is referenced by:  mbfi1fseqlem3  23484  mbfi1fseqlem4  23485  mbfi1fseqlem5  23486  mbfi1fseqlem6  23487