This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem itgeq1fOLD

Description: Obsolete version of itgeq1f as of 1-Sep-2025. (Contributed by Mario Carneiro, 28-Jun-2014) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses itgeq1f.1 
|- F/_ x A
itgeq1f.2 
|- F/_ x B
Assertion itgeq1fOLD 
|- ( A = B -> S. A C _d x = S. B C _d x )

Proof

Step Hyp Ref Expression
1 itgeq1f.1 
 |-  F/_ x A
2 itgeq1f.2 
 |-  F/_ x B
3 eqid 
 |-  RR = RR
4 1 2 nfeq 
 |-  F/ x A = B
5 eleq2 
 |-  ( A = B -> ( x e. A <-> x e. B ) )
6 5 anbi1d 
 |-  ( A = B -> ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) <-> ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) ) )
7 6 ifbid 
 |-  ( A = B -> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) = if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) )
8 7 a1d 
 |-  ( A = B -> ( x e. RR -> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) = if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) )
9 4 8 ralrimi 
 |-  ( A = B -> A. x e. RR if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) = if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) )
10 mpteq12 
 |-  ( ( RR = RR /\ A. x e. RR if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) = if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) -> ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) = ( x e. RR |-> if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) )
11 3 9 10 sylancr 
 |-  ( A = B -> ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) = ( x e. RR |-> if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) )
12 11 fveq2d 
 |-  ( A = B -> ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) = ( S.2 ` ( x e. RR |-> if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) )
13 12 oveq2d 
 |-  ( A = B -> ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) ) = ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) ) )
14 13 sumeq2sdv 
 |-  ( A = B -> sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) ) = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) ) )
15 eqid 
 |-  ( Re ` ( C / ( _i ^ k ) ) ) = ( Re ` ( C / ( _i ^ k ) ) )
16 15 dfitg 
 |-  S. A C _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) )
17 15 dfitg 
 |-  S. B C _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. B /\ 0 <_ ( Re ` ( C / ( _i ^ k ) ) ) ) , ( Re ` ( C / ( _i ^ k ) ) ) , 0 ) ) ) )
18 14 16 17 3eqtr4g 
 |-  ( A = B -> S. A C _d x = S. B C _d x )