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Theorem fsum0diag2 12493
Description: Two ways to express "the sum of  A ( j ,  k ) over the triangular region  0  <_  j, 
0  <_  k,  j  +  k  <_  N." (Contributed by Mario Carneiro, 21-Jul-2014.)
Hypotheses
Ref Expression
fsum0diag2.1  |-  ( x  =  k  ->  B  =  A )
fsum0diag2.2  |-  ( x  =  ( k  -  j )  ->  B  =  C )
fsum0diag2.3  |-  ( (
ph  /\  ( j  e.  ( 0 ... N
)  /\  k  e.  ( 0 ... ( N  -  j )
) ) )  ->  A  e.  CC )
Assertion
Ref Expression
fsum0diag2  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N )
sum_ k  e.  ( 0 ... ( N  -  j ) ) A  =  sum_ k  e.  ( 0 ... N
) sum_ j  e.  ( 0 ... k ) C )
Distinct variable groups:    j, k, x, N    ph, j, k    B, k    x, A    x, C
Allowed substitution hints:    ph( x)    A( j, k)    B( x, j)    C( j, k)

Proof of Theorem fsum0diag2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 fznn0sub2 11018 . . . . . . 7  |-  ( n  e.  ( 0 ... ( N  -  j
) )  ->  (
( N  -  j
)  -  n )  e.  ( 0 ... ( N  -  j
) ) )
21ad2antll 710 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( 0 ... N
)  /\  n  e.  ( 0 ... ( N  -  j )
) ) )  -> 
( ( N  -  j )  -  n
)  e.  ( 0 ... ( N  -  j ) ) )
3 fsum0diag2.3 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( 0 ... N
)  /\  k  e.  ( 0 ... ( N  -  j )
) ) )  ->  A  e.  CC )
43expr 599 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
k  e.  ( 0 ... ( N  -  j ) )  ->  A  e.  CC )
)
54ralrimiv 2731 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  A. k  e.  ( 0 ... ( N  -  j )
) A  e.  CC )
6 fsum0diag2.1 . . . . . . . . . 10  |-  ( x  =  k  ->  B  =  A )
76eleq1d 2453 . . . . . . . . 9  |-  ( x  =  k  ->  ( B  e.  CC  <->  A  e.  CC ) )
87cbvralv 2875 . . . . . . . 8  |-  ( A. x  e.  ( 0 ... ( N  -  j ) ) B  e.  CC  <->  A. k  e.  ( 0 ... ( N  -  j )
) A  e.  CC )
95, 8sylibr 204 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  A. x  e.  ( 0 ... ( N  -  j )
) B  e.  CC )
109adantrr 698 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( 0 ... N
)  /\  n  e.  ( 0 ... ( N  -  j )
) ) )  ->  A. x  e.  (
0 ... ( N  -  j ) ) B  e.  CC )
11 nfcsb1v 3226 . . . . . . . 8  |-  F/_ x [_ ( ( N  -  j )  -  n
)  /  x ]_ B
1211nfel1 2533 . . . . . . 7  |-  F/ x [_ ( ( N  -  j )  -  n
)  /  x ]_ B  e.  CC
13 csbeq1a 3202 . . . . . . . 8  |-  ( x  =  ( ( N  -  j )  -  n )  ->  B  =  [_ ( ( N  -  j )  -  n )  /  x ]_ B )
1413eleq1d 2453 . . . . . . 7  |-  ( x  =  ( ( N  -  j )  -  n )  ->  ( B  e.  CC  <->  [_ ( ( N  -  j )  -  n )  /  x ]_ B  e.  CC ) )
1512, 14rspc 2989 . . . . . 6  |-  ( ( ( N  -  j
)  -  n )  e.  ( 0 ... ( N  -  j
) )  ->  ( A. x  e.  (
0 ... ( N  -  j ) ) B  e.  CC  ->  [_ (
( N  -  j
)  -  n )  /  x ]_ B  e.  CC ) )
162, 10, 15sylc 58 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( 0 ... N
)  /\  n  e.  ( 0 ... ( N  -  j )
) ) )  ->  [_ ( ( N  -  j )  -  n
)  /  x ]_ B  e.  CC )
1716fsum0diag 12488 . . . 4  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N )
sum_ n  e.  (
0 ... ( N  -  j ) ) [_ ( ( N  -  j )  -  n
)  /  x ]_ B  =  sum_ n  e.  ( 0 ... N
) sum_ j  e.  ( 0 ... ( N  -  n ) )
[_ ( ( N  -  j )  -  n )  /  x ]_ B )
18 nfcsb1v 3226 . . . . . . . . . 10  |-  F/_ x [_ k  /  x ]_ B
1918nfel1 2533 . . . . . . . . 9  |-  F/ x [_ k  /  x ]_ B  e.  CC
20 csbeq1a 3202 . . . . . . . . . 10  |-  ( x  =  k  ->  B  =  [_ k  /  x ]_ B )
2120eleq1d 2453 . . . . . . . . 9  |-  ( x  =  k  ->  ( B  e.  CC  <->  [_ k  /  x ]_ B  e.  CC ) )
2219, 21rspc 2989 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  ( A. x  e.  (
0 ... ( N  -  j ) ) B  e.  CC  ->  [_ k  /  x ]_ B  e.  CC ) )
239, 22mpan9 456 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  k  e.  ( 0 ... ( N  -  j )
) )  ->  [_ k  /  x ]_ B  e.  CC )
24 csbeq1 3197 . . . . . . 7  |-  ( k  =  ( ( 0  +  ( N  -  j ) )  -  n )  ->  [_ k  /  x ]_ B  = 
[_ ( ( 0  +  ( N  -  j ) )  -  n )  /  x ]_ B )
2523, 24fsumrev2 12492 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  sum_ k  e.  ( 0 ... ( N  -  j )
) [_ k  /  x ]_ B  =  sum_ n  e.  ( 0 ... ( N  -  j
) ) [_ (
( 0  +  ( N  -  j ) )  -  n )  /  x ]_ B
)
26 elfz3nn0 11016 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... N )  ->  N  e.  NN0 )
2726ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  n  e.  ( 0 ... ( N  -  j )
) )  ->  N  e.  NN0 )
28 elfzelz 10991 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ZZ )
2928ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  n  e.  ( 0 ... ( N  -  j )
) )  ->  j  e.  ZZ )
30 nn0cn 10163 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e.  CC )
31 zcn 10219 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  CC )
32 subcl 9237 . . . . . . . . . . . 12  |-  ( ( N  e.  CC  /\  j  e.  CC )  ->  ( N  -  j
)  e.  CC )
3330, 31, 32syl2an 464 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  j  e.  ZZ )  ->  ( N  -  j
)  e.  CC )
3427, 29, 33syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  n  e.  ( 0 ... ( N  -  j )
) )  ->  ( N  -  j )  e.  CC )
35 addid2 9181 . . . . . . . . . 10  |-  ( ( N  -  j )  e.  CC  ->  (
0  +  ( N  -  j ) )  =  ( N  -  j ) )
3634, 35syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  n  e.  ( 0 ... ( N  -  j )
) )  ->  (
0  +  ( N  -  j ) )  =  ( N  -  j ) )
3736oveq1d 6035 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  n  e.  ( 0 ... ( N  -  j )
) )  ->  (
( 0  +  ( N  -  j ) )  -  n )  =  ( ( N  -  j )  -  n ) )
3837csbeq1d 3200 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( 0 ... N
) )  /\  n  e.  ( 0 ... ( N  -  j )
) )  ->  [_ (
( 0  +  ( N  -  j ) )  -  n )  /  x ]_ B  =  [_ ( ( N  -  j )  -  n )  /  x ]_ B )
3938sumeq2dv 12424 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  sum_ n  e.  ( 0 ... ( N  -  j )
) [_ ( ( 0  +  ( N  -  j ) )  -  n )  /  x ]_ B  =  sum_ n  e.  ( 0 ... ( N  -  j
) ) [_ (
( N  -  j
)  -  n )  /  x ]_ B
)
4025, 39eqtrd 2419 . . . . 5  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  sum_ k  e.  ( 0 ... ( N  -  j )
) [_ k  /  x ]_ B  =  sum_ n  e.  ( 0 ... ( N  -  j
) ) [_ (
( N  -  j
)  -  n )  /  x ]_ B
)
4140sumeq2dv 12424 . . . 4  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N )
sum_ k  e.  ( 0 ... ( N  -  j ) )
[_ k  /  x ]_ B  =  sum_ j  e.  ( 0 ... N ) sum_ n  e.  ( 0 ... ( N  -  j
) ) [_ (
( N  -  j
)  -  n )  /  x ]_ B
)
42 elfz3nn0 11016 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... N )  ->  N  e.  NN0 )
4342adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 0 ... N
) )  ->  N  e.  NN0 )
44 addid2 9181 . . . . . . . . 9  |-  ( N  e.  CC  ->  (
0  +  N )  =  N )
4543, 30, 443syl 19 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... N
) )  ->  (
0  +  N )  =  N )
4645oveq1d 6035 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 0 ... N
) )  ->  (
( 0  +  N
)  -  n )  =  ( N  -  n ) )
4746oveq2d 6036 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 0 ... N
) )  ->  (
0 ... ( ( 0  +  N )  -  n ) )  =  ( 0 ... ( N  -  n )
) )
4846oveq1d 6035 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 0 ... N
) )  ->  (
( ( 0  +  N )  -  n
)  -  j )  =  ( ( N  -  n )  -  j ) )
4948adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  (
( ( 0  +  N )  -  n
)  -  j )  =  ( ( N  -  n )  -  j ) )
5042ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  N  e.  NN0 )
51 elfzelz 10991 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... N )  ->  n  e.  ZZ )
5251ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  n  e.  ZZ )
53 elfzelz 10991 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... ( N  -  n
) )  ->  j  e.  ZZ )
5453adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  j  e.  ZZ )
55 zcn 10219 . . . . . . . . . 10  |-  ( n  e.  ZZ  ->  n  e.  CC )
56 sub32 9267 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  n  e.  CC  /\  j  e.  CC )  ->  (
( N  -  n
)  -  j )  =  ( ( N  -  j )  -  n ) )
5730, 55, 31, 56syl3an 1226 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  n  e.  ZZ  /\  j  e.  ZZ )  ->  (
( N  -  n
)  -  j )  =  ( ( N  -  j )  -  n ) )
5850, 52, 54, 57syl3anc 1184 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  (
( N  -  n
)  -  j )  =  ( ( N  -  j )  -  n ) )
5949, 58eqtrd 2419 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  (
( ( 0  +  N )  -  n
)  -  j )  =  ( ( N  -  j )  -  n ) )
6059csbeq1d 3200 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... ( N  -  n )
) )  ->  [_ (
( ( 0  +  N )  -  n
)  -  j )  /  x ]_ B  =  [_ ( ( N  -  j )  -  n )  /  x ]_ B )
6147, 60sumeq12rdv 12428 . . . . 5  |-  ( (
ph  /\  n  e.  ( 0 ... N
) )  ->  sum_ j  e.  ( 0 ... (
( 0  +  N
)  -  n ) ) [_ ( ( ( 0  +  N
)  -  n )  -  j )  /  x ]_ B  =  sum_ j  e.  ( 0 ... ( N  -  n ) ) [_ ( ( N  -  j )  -  n
)  /  x ]_ B )
6261sumeq2dv 12424 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
0 ... N ) sum_ j  e.  ( 0 ... ( ( 0  +  N )  -  n ) ) [_ ( ( ( 0  +  N )  -  n )  -  j
)  /  x ]_ B  =  sum_ n  e.  ( 0 ... N
) sum_ j  e.  ( 0 ... ( N  -  n ) )
[_ ( ( N  -  j )  -  n )  /  x ]_ B )
6317, 41, 623eqtr4d 2429 . . 3  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N )
sum_ k  e.  ( 0 ... ( N  -  j ) )
[_ k  /  x ]_ B  =  sum_ n  e.  ( 0 ... N ) sum_ j  e.  ( 0 ... (
( 0  +  N
)  -  n ) ) [_ ( ( ( 0  +  N
)  -  n )  -  j )  /  x ]_ B )
64 fzfid 11239 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
0 ... k )  e. 
Fin )
65 elfzuz3 10988 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... k )  ->  k  e.  ( ZZ>= `  j )
)
6665adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  k  e.  ( ZZ>= `  j )
)
67 elfzuz3 10988 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  k )
)
6867adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  k )
)
6968adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  N  e.  ( ZZ>= `  k )
)
70 elfzuzb 10985 . . . . . . . . 9  |-  ( k  e.  ( j ... N )  <->  ( k  e.  ( ZZ>= `  j )  /\  N  e.  ( ZZ>=
`  k ) ) )
7166, 69, 70sylanbrc 646 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  k  e.  ( j ... N
) )
72 elfzelz 10991 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... k )  ->  j  e.  ZZ )
7372adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  j  e.  ZZ )
74 elfzel2 10989 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  N  e.  ZZ )
7574ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  N  e.  ZZ )
76 elfzelz 10991 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  k  e.  ZZ )
7776ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  k  e.  ZZ )
78 fzsubel 11020 . . . . . . . . 9  |-  ( ( ( j  e.  ZZ  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( k  e.  ( j ... N )  <-> 
( k  -  j
)  e.  ( ( j  -  j ) ... ( N  -  j ) ) ) )
7973, 75, 77, 73, 78syl22anc 1185 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  (
k  e.  ( j ... N )  <->  ( k  -  j )  e.  ( ( j  -  j ) ... ( N  -  j )
) ) )
8071, 79mpbid 202 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  (
k  -  j )  e.  ( ( j  -  j ) ... ( N  -  j
) ) )
81 subid 9253 . . . . . . . . 9  |-  ( j  e.  CC  ->  (
j  -  j )  =  0 )
8273, 31, 813syl 19 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  (
j  -  j )  =  0 )
8382oveq1d 6035 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  (
( j  -  j
) ... ( N  -  j ) )  =  ( 0 ... ( N  -  j )
) )
8480, 83eleqtrd 2463 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  (
k  -  j )  e.  ( 0 ... ( N  -  j
) ) )
85 simpll 731 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  ph )
86 fzss2 11024 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  k
)  ->  ( 0 ... k )  C_  ( 0 ... N
) )
8768, 86syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
0 ... k )  C_  ( 0 ... N
) )
8887sselda 3291 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  j  e.  ( 0 ... N
) )
8985, 88, 9syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  A. x  e.  ( 0 ... ( N  -  j )
) B  e.  CC )
90 nfcsb1v 3226 . . . . . . . 8  |-  F/_ x [_ ( k  -  j
)  /  x ]_ B
9190nfel1 2533 . . . . . . 7  |-  F/ x [_ ( k  -  j
)  /  x ]_ B  e.  CC
92 csbeq1a 3202 . . . . . . . 8  |-  ( x  =  ( k  -  j )  ->  B  =  [_ ( k  -  j )  /  x ]_ B )
9392eleq1d 2453 . . . . . . 7  |-  ( x  =  ( k  -  j )  ->  ( B  e.  CC  <->  [_ ( k  -  j )  /  x ]_ B  e.  CC ) )
9491, 93rspc 2989 . . . . . 6  |-  ( ( k  -  j )  e.  ( 0 ... ( N  -  j
) )  ->  ( A. x  e.  (
0 ... ( N  -  j ) ) B  e.  CC  ->  [_ (
k  -  j )  /  x ]_ B  e.  CC ) )
9584, 89, 94sylc 58 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  j  e.  ( 0 ... k
) )  ->  [_ (
k  -  j )  /  x ]_ B  e.  CC )
9664, 95fsumcl 12454 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  sum_ j  e.  ( 0 ... k
) [_ ( k  -  j )  /  x ]_ B  e.  CC )
97 oveq2 6028 . . . . 5  |-  ( k  =  ( ( 0  +  N )  -  n )  ->  (
0 ... k )  =  ( 0 ... (
( 0  +  N
)  -  n ) ) )
98 oveq1 6027 . . . . . . 7  |-  ( k  =  ( ( 0  +  N )  -  n )  ->  (
k  -  j )  =  ( ( ( 0  +  N )  -  n )  -  j ) )
9998csbeq1d 3200 . . . . . 6  |-  ( k  =  ( ( 0  +  N )  -  n )  ->  [_ (
k  -  j )  /  x ]_ B  =  [_ ( ( ( 0  +  N )  -  n )  -  j )  /  x ]_ B )
10099adantr 452 . . . . 5  |-  ( ( k  =  ( ( 0  +  N )  -  n )  /\  j  e.  ( 0 ... k ) )  ->  [_ ( k  -  j )  /  x ]_ B  =  [_ (
( ( 0  +  N )  -  n
)  -  j )  /  x ]_ B
)
10197, 100sumeq12dv 12427 . . . 4  |-  ( k  =  ( ( 0  +  N )  -  n )  ->  sum_ j  e.  ( 0 ... k
) [_ ( k  -  j )  /  x ]_ B  =  sum_ j  e.  ( 0 ... ( ( 0  +  N )  -  n ) ) [_ ( ( ( 0  +  N )  -  n )  -  j
)  /  x ]_ B )
10296, 101fsumrev2 12492 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N )
sum_ j  e.  ( 0 ... k )
[_ ( k  -  j )  /  x ]_ B  =  sum_ n  e.  ( 0 ... N ) sum_ j  e.  ( 0 ... (
( 0  +  N
)  -  n ) ) [_ ( ( ( 0  +  N
)  -  n )  -  j )  /  x ]_ B )
10363, 102eqtr4d 2422 . 2  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N )
sum_ k  e.  ( 0 ... ( N  -  j ) )
[_ k  /  x ]_ B  =  sum_ k  e.  ( 0 ... N ) sum_ j  e.  ( 0 ... k ) [_ ( k  -  j
)  /  x ]_ B )
104 vex 2902 . . . . . 6  |-  k  e. 
_V
105 nfcv 2523 . . . . . 6  |-  F/_ x A
106104, 105, 6csbief 3235 . . . . 5  |-  [_ k  /  x ]_ B  =  A
107106a1i 11 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  [_ k  /  x ]_ B  =  A
)
108107sumeq2dv 12424 . . 3  |-  ( j  e.  ( 0 ... N )  ->  sum_ k  e.  ( 0 ... ( N  -  j )
) [_ k  /  x ]_ B  =  sum_ k  e.  ( 0 ... ( N  -  j ) ) A )
109108sumeq2i 12420 . 2  |-  sum_ j  e.  ( 0 ... N
) sum_ k  e.  ( 0 ... ( N  -  j ) )
[_ k  /  x ]_ B  =  sum_ j  e.  ( 0 ... N ) sum_ k  e.  ( 0 ... ( N  -  j ) ) A
110 ovex 6045 . . . . . 6  |-  ( k  -  j )  e. 
_V
111 nfcv 2523 . . . . . 6  |-  F/_ x C
112 fsum0diag2.2 . . . . . 6  |-  ( x  =  ( k  -  j )  ->  B  =  C )
113110, 111, 112csbief 3235 . . . . 5  |-  [_ (
k  -  j )  /  x ]_ B  =  C
114113a1i 11 . . . 4  |-  ( ( k  e.  ( 0 ... N )  /\  j  e.  ( 0 ... k ) )  ->  [_ ( k  -  j )  /  x ]_ B  =  C
)
115114sumeq2dv 12424 . . 3  |-  ( k  e.  ( 0 ... N )  ->  sum_ j  e.  ( 0 ... k
) [_ ( k  -  j )  /  x ]_ B  =  sum_ j  e.  ( 0 ... k ) C )
116115sumeq2i 12420 . 2  |-  sum_ k  e.  ( 0 ... N
) sum_ j  e.  ( 0 ... k )
[_ ( k  -  j )  /  x ]_ B  =  sum_ k  e.  ( 0 ... N ) sum_ j  e.  ( 0 ... k ) C
117103, 109, 1163eqtr3g 2442 1  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N )
sum_ k  e.  ( 0 ... ( N  -  j ) ) A  =  sum_ k  e.  ( 0 ... N
) sum_ j  e.  ( 0 ... k ) C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   [_csb 3194    C_ wss 3263   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923    + caddc 8926    - cmin 9223   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420   ...cfz 10975   sum_csu 12406
This theorem is referenced by:  mertens  12590  plymullem1  20000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407
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