Theorem nfsum1 12019

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

Step	Hyp	Ref	Expression
1		df-sumdc 12017	. 2 |- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ 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 |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
2		nfcv 2384	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2384	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3230	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2378	. . . . . . . 8 |- F/ k j e. A
7	6	nfdc 1707	. . . . . . 7 |- F/ kDECID j e. A
8	4, 7	nfralxy 2580	. . . . . 6 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2384	. . . . . . . 8 |- F/_ k m
10		nfcv 2384	. . . . . . . 8 |- F/_ k +
11	3	nfcri 2378	. . . . . . . . . 10 |- F/ k n e. A
12		nfcsb1v 3170	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
13		nfcv 2384	. . . . . . . . . 10 |- F/_ k 0
14	11, 12, 13	nfif 3647	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
15	2, 14	nfmpt 4195	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
16	9, 10, 15	nfseq 10805	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
17		nfcv 2384	. . . . . . 7 |- F/_ k ~~>
18		nfcv 2384	. . . . . . 7 |- F/_ k x
19	16, 17, 18	nfbr 4149	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
20	5, 8, 19	nf3an 1615	. . . . 5 |- F/ k ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x )
21	2, 20	nfrexya 2583	. . . 4 |- F/ k E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x )
22		nfcv 2384	. . . . 5 |- F/_ k NN
23		nfcv 2384	. . . . . . . 8 |- F/_ k f
24		nfcv 2384	. . . . . . . 8 |- F/_ k ( 1 ... m )
25	23, 24, 3	nff1o 5603	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
26		nfcv 2384	. . . . . . . . . 10 |- F/_ k 1
27		nfv 1577	. . . . . . . . . . . 12 |- F/ k n <_ m
28		nfcsb1v 3170	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
29	27, 28, 13	nfif 3647	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
30	22, 29	nfmpt 4195	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
31	26, 10, 30	nfseq 10805	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
32	31, 9	nffv 5671	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
33	32	nfeq2 2396	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
34	25, 33	nfan 1614	. . . . . 6 |- F/ k ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
35	34	nfex 1686	. . . . 5 |- F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
36	22, 35	nfrexya 2583	. . . 4 |- F/ k E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
37	21, 36	nfor 1623	. . 3 |- F/ k ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ 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 |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) )
38	37	nfiotaw 5307	. 2 |- F/_ k ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ 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 |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
39	1, 38	nfcxfr 2381	1 |- F/_ k sum_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   F/_wnfc 2371   A.wral 2520   E.wrex 2521   [_csb 3137   C_ wss 3210   ifcif 3616   class class class wbr 4102   |-> cmpt 4164   iotacio 5301   -1-1-onto->wf1o 5342   ` cfv 5343  (class class class)co 6041   0cc0 8115   1c1 8116   + caddc 8118   <_ cle 8297   NNcn 9225   ZZcz 9563   ZZ>=cuz 9839   ...cfz 10328   seqcseq 10795   ~~> cli 11941   sum_csu 12016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-if 3617  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-opab 4165  df-mpt 4166  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-f1 5348  df-fo 5349  df-f1o 5350  df-fv 5351  df-ov 6044  df-oprab 6045  df-mpo 6046  df-recs 6527  df-frec 6613  df-seqfrec 10796  df-sumdc 12017

This theorem is referenced by:  mertenslem2  12200