Theorem sumeq1 10192

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 3020	. . . . . 6 |- ( A = B -> ( A C_ ( ZZ>= ` m ) <-> B C_ ( ZZ>= ` m ) ) )
2		simpl 107	. . . . . . . . . . 11 |- ( ( A = B /\ n e. ZZ ) -> A = B )
3	2	eleq2d 2148	. . . . . . . . . 10 |- ( ( A = B /\ n e. ZZ ) -> ( n e. A <-> n e. B ) )
4	3	ifbid 3370	. . . . . . . . 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 3868	. . . . . . . 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 9436	. . . . . . . 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 3795	. . . . . 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 456	. . . . 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 2369	. . . 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 5139	. . . . . . 7 |- ( A = B -> ( f : ( 1 ... m ) -1-1-onto-> A <-> f : ( 1 ... m ) -1-1-onto-> B ) )
12	11	anbi1d 452	. . . . . 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 1746	. . . . 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 2369	. . . 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 739	. . 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 4908	. 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 10191	. 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 10191	. 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 2138	1 |- ( A = B -> sum_ k e. A C = sum_ k e. B C )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   E.wex 1421   e. wcel 1433   E.wrex 2349   [_csb 2908   C_ wss 2973   ifcif 3351   class class class wbr 3785   |-> cmpt 3839   iotacio 4885   -1-1-onto->wf1o 4921   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   NNcn 8039   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029   seqcseq 9431   ~~> cli 10117   sum_csu 10190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-if 3352  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-recs 5943  df-frec 6001  df-iseq 9432  df-sum 10191

This theorem is referenced by: (None)