Theorem sumeq1 9874

Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Assertion

Ref	Expression
sumeq1	|- ( A = B -> sum_ k e. A C = sum_ k e. B C )

Proof of Theorem sumeq1

Dummy variables f m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 2966	. . . . . 6 |- ( A = B -> ( A C_ ( ZZ>= ` m ) <-> B C_ ( ZZ>= ` m ) ) )
2		simpl 102	. . . . . . . . . . 11 |- ( ( A = B /\ n e. ZZ ) -> A = B )
3	2	eleq2d 2107	. . . . . . . . . 10 |- ( ( A = B /\ n e. ZZ ) -> ( n e. A <-> n e. B ) )
4	3	ifbid 3349	. . . . . . . . 9 |- ( ( A = B /\ n e. ZZ ) -> if ( n e. A , [_ n / k ]_ C , 0 ) = if ( n e. B , [_ n / k ]_ C , 0 ) )
5	4	mpteq2dva 3847	. . . . . . . 8 |- ( A = B -> ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) = ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) )
6		iseqeq3 9216	. . . . . . . 8 |- ( ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) = ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) -> seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) = seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) )
7	5, 6	syl 14	. . . . . . 7 |- ( A = B -> seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) = seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) )
8	7	breq1d 3774	. . . . . 6 |- ( A = B -> ( seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x <-> seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) )
9	1, 8	anbi12d 442	. . . . 5 |- ( A = B -> ( ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) <-> ( B C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) ) )
10	9	rexbidv 2327	. . . 4 |- ( A = B -> ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) <-> E. m e. ZZ ( B C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) ) )
11		f1oeq3 5119	. . . . . . 7 |- ( A = B -> ( f : ( 1 ... m ) -1-1-onto-> A <-> f : ( 1 ... m ) -1-1-onto-> B ) )
12	11	anbi1d 438	. . . . . 6 |- ( A = B -> ( ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) <-> ( f : ( 1 ... m ) -1-1-onto-> B /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) )
13	12	exbidv 1706	. . . . 5 |- ( A = B -> ( E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) <-> E. f ( f : ( 1 ... m ) -1-1-onto-> B /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) )
14	13	rexbidv 2327	. . . 4 |- ( A = B -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) <-> E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> B /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) )
15	10, 14	orbi12d 707	. . 3 |- ( A = B -> ( ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) <-> ( E. m e. ZZ ( B C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> B /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) ) )
16	15	iotabidv 4888	. 2 |- ( A = B -> ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) ) = ( iota x ( E. m e. ZZ ( B C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> B /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) ) )
17		df-sum 9873	. 2 |- sum_ k e. A C = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) )
18		df-sum 9873	. 2 |- sum_ k e. B C = ( iota x ( E. m e. ZZ ( B C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. B , [_ n / k ]_ C , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> B /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ C ) , CC ) ` m ) ) ) )
19	16, 17, 18	3eqtr4g 2097	1 |- ( A = B -> sum_ k e. A C = sum_ k e. B C )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307   [_csb 2852   C_ wss 2917   ifcif 3331   class class class wbr 3764   |-> cmpt 3818   iotacio 4865   -1-1-onto->wf1o 4901   ` cfv 4902  (class class class)co 5512   CCcc 6887   0cc0 6889   1c1 6890   + caddc 6892   NNcn 7914   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874   seqcseq 9211   ~~> cli 9799   sum_csu 9872

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212  df-sum 9873

This theorem is referenced by: (None)