Theorem nfsum1 14420

Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypothesis

Ref	Expression
nfsum1.1	|- F/_ k A

Assertion

Ref	Expression
nfsum1	|- F/_ k sum_ k e. A B

Proof of Theorem nfsum1

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

Step	Hyp	Ref	Expression
1		df-sum 14417	. 2 |- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
2		nfcv 2764	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2764	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3596	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6		nfcv 2764	. . . . . . . 8 |- F/_ k m
7		nfcv 2764	. . . . . . . 8 |- F/_ k +
8	3	nfcri 2758	. . . . . . . . . 10 |- F/ k n e. A
9		nfcsb1v 3549	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
10		nfcv 2764	. . . . . . . . . 10 |- F/_ k 0
11	8, 9, 10	nfif 4115	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
12	2, 11	nfmpt 4746	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
13	6, 7, 12	nfseq 12811	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
14		nfcv 2764	. . . . . . 7 |- F/_ k ~~>
15		nfcv 2764	. . . . . . 7 |- F/_ k x
16	13, 14, 15	nfbr 4699	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
17	5, 16	nfan 1828	. . . . 5 |- F/ k ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x )
18	2, 17	nfrex 3007	. . . 4 |- F/ k E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x )
19		nfcv 2764	. . . . 5 |- F/_ k NN
20		nfcv 2764	. . . . . . . 8 |- F/_ k f
21		nfcv 2764	. . . . . . . 8 |- F/_ k ( 1 ... m )
22	20, 21, 3	nff1o 6135	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
23		nfcv 2764	. . . . . . . . . 10 |- F/_ k 1
24		nfcsb1v 3549	. . . . . . . . . . 11 |- F/_ k [_ ( f ` n ) / k ]_ B
25	19, 24	nfmpt 4746	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
26	23, 7, 25	nfseq 12811	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) )
27	26, 6	nffv 6198	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m )
28	27	nfeq2 2780	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m )
29	22, 28	nfan 1828	. . . . . 6 |- F/ k ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
30	29	nfex 2154	. . . . 5 |- F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
31	19, 30	nfrex 3007	. . . 4 |- F/ k E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
32	18, 31	nfor 1834	. . 3 |- F/ k ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) )
33	32	nfiota 5855	. 2 |- F/_ k ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
34	1, 33	nfcxfr 2762	1 |- F/_ k sum_ k e. A B

Syntax hints:   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   E.wrex 2913   [_csb 3533   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   iotacio 5849   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   sum_csu 14416

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-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-br 4654  df-opab 4713  df-mpt 4730  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-pred 5680  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-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-sum 14417

This theorem is referenced by:  dvmptfprod  40160  dvnprodlem1  40161  fourierdlem112  40435  etransclem32  40483  sge0reuz  40664