Theorem itgvallem 23551

Description: Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypothesis

Ref	Expression
itgvallem.1	|- ( _i ^ K ) = T

Assertion

Ref	Expression
itgvallem	|- ( k = K -> ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( B / ( _i ^ k ) ) ) ) , ( Re ` ( B / ( _i ^ k ) ) ) , 0 ) ) ) = ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( B / T ) ) ) , ( Re ` ( B / T ) ) , 0 ) ) ) )

Distinct variable groups:   x, k   x, K

Allowed substitution hints:   A( x, k)   B( x, k)   T( x, k)   K( k)

Proof of Theorem itgvallem

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . . 9 |- ( k = K -> ( _i ^ k ) = ( _i ^ K ) )
2		itgvallem.1	. . . . . . . . 9 |- ( _i ^ K ) = T
3	1, 2	syl6eq 2672	. . . . . . . 8 |- ( k = K -> ( _i ^ k ) = T )
4	3	oveq2d 6666	. . . . . . 7 |- ( k = K -> ( B / ( _i ^ k ) ) = ( B / T ) )
5	4	fveq2d 6195	. . . . . 6 |- ( k = K -> ( Re ` ( B / ( _i ^ k ) ) ) = ( Re ` ( B / T ) ) )
6	5	breq2d 4665	. . . . 5 |- ( k = K -> ( 0 <_ ( Re ` ( B / ( _i ^ k ) ) ) <-> 0 <_ ( Re ` ( B / T ) ) ) )
7	6	anbi2d 740	. . . 4 |- ( k = K -> ( ( x e. A /\ 0 <_ ( Re ` ( B / ( _i ^ k ) ) ) ) <-> ( x e. A /\ 0 <_ ( Re ` ( B / T ) ) ) ) )
8	7, 5	ifbieq1d 4109	. . 3 |- ( k = K -> if ( ( x e. A /\ 0 <_ ( Re ` ( B / ( _i ^ k ) ) ) ) , ( Re ` ( B / ( _i ^ k ) ) ) , 0 ) = if ( ( x e. A /\ 0 <_ ( Re ` ( B / T ) ) ) , ( Re ` ( B / T ) ) , 0 ) )
9	8	mpteq2dv 4745	. 2 |- ( k = K -> ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( B / ( _i ^ k ) ) ) ) , ( Re ` ( B / ( _i ^ k ) ) ) , 0 ) ) = ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( B / T ) ) ) , ( Re ` ( B / T ) ) , 0 ) ) )
10	9	fveq2d 6195	1 |- ( k = K -> ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( B / ( _i ^ k ) ) ) ) , ( Re ` ( B / ( _i ^ k ) ) ) , 0 ) ) ) = ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( B / T ) ) ) , ( Re ` ( B / T ) ) , 0 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   _ici 9938   <_ cle 10075   / cdiv 10684   ^cexp 12860   Recre 13837   S.2citg2 23385

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  iblcnlem1  23554  itgcnlem  23556