Step |
Hyp |
Ref |
Expression |
1 |
|
nfsum.1 |
|- F/_ x A |
2 |
|
nfsum.2 |
|- F/_ x B |
3 |
|
df-sum |
|- sum_ k e. A B = ( iota z ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) ) |
4 |
|
nfcv |
|- F/_ x ZZ |
5 |
|
nfcv |
|- F/_ x ( ZZ>= ` m ) |
6 |
1 5
|
nfss |
|- F/ x A C_ ( ZZ>= ` m ) |
7 |
|
nfcv |
|- F/_ x m |
8 |
|
nfcv |
|- F/_ x + |
9 |
1
|
nfcri |
|- F/ x n e. A |
10 |
|
nfcv |
|- F/_ x n |
11 |
10 2
|
nfcsbw |
|- F/_ x [_ n / k ]_ B |
12 |
|
nfcv |
|- F/_ x 0 |
13 |
9 11 12
|
nfif |
|- F/_ x if ( n e. A , [_ n / k ]_ B , 0 ) |
14 |
4 13
|
nfmpt |
|- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) |
15 |
7 8 14
|
nfseq |
|- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) |
16 |
|
nfcv |
|- F/_ x ~~> |
17 |
|
nfcv |
|- F/_ x z |
18 |
15 16 17
|
nfbr |
|- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z |
19 |
6 18
|
nfan |
|- F/ x ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z ) |
20 |
4 19
|
nfrex |
|- F/ x E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z ) |
21 |
|
nfcv |
|- F/_ x NN |
22 |
|
nfcv |
|- F/_ x f |
23 |
|
nfcv |
|- F/_ x ( 1 ... m ) |
24 |
22 23 1
|
nff1o |
|- F/ x f : ( 1 ... m ) -1-1-onto-> A |
25 |
|
nfcv |
|- F/_ x 1 |
26 |
|
nfcv |
|- F/_ x ( f ` n ) |
27 |
26 2
|
nfcsbw |
|- F/_ x [_ ( f ` n ) / k ]_ B |
28 |
21 27
|
nfmpt |
|- F/_ x ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) |
29 |
25 8 28
|
nfseq |
|- F/_ x seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) |
30 |
29 7
|
nffv |
|- F/_ x ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) |
31 |
30
|
nfeq2 |
|- F/ x z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) |
32 |
24 31
|
nfan |
|- F/ x ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) |
33 |
32
|
nfex |
|- F/ x E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) |
34 |
21 33
|
nfrex |
|- F/ x E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) |
35 |
20 34
|
nfor |
|- F/ x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) |
36 |
35
|
nfiotaw |
|- F/_ x ( iota z ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) ) |
37 |
3 36
|
nfcxfr |
|- F/_ x sum_ k e. A B |