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Theorem seqcoll2 12516
Description: The function  F contains a sparse set of non-zero values to be summed. The function  G is an order isomorphism from the set of non-zero values of  F to a 1-based finite sequence, and  H collects these non-zero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
seqcoll2.1  |-  ( (
ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )
seqcoll2.1b  |-  ( (
ph  /\  k  e.  S )  ->  (
k  .+  Z )  =  k )
seqcoll2.c  |-  ( (
ph  /\  ( k  e.  S  /\  n  e.  S ) )  -> 
( k  .+  n
)  e.  S )
seqcoll2.a  |-  ( ph  ->  Z  e.  S )
seqcoll2.2  |-  ( ph  ->  G  Isom  <  ,  <  ( ( 1 ... ( # `
 A ) ) ,  A ) )
seqcoll2.3  |-  ( ph  ->  A  =/=  (/) )
seqcoll2.5  |-  ( ph  ->  A  C_  ( M ... N ) )
seqcoll2.6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  S
)
seqcoll2.7  |-  ( (
ph  /\  k  e.  ( ( M ... N )  \  A
) )  ->  ( F `  k )  =  Z )
seqcoll2.8  |-  ( (
ph  /\  n  e.  ( 1 ... ( # `
 A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
 n ) ) )
Assertion
Ref Expression
seqcoll2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq 1 ( 
.+  ,  H ) `
 ( # `  A
) ) )
Distinct variable groups:    k, n, A    k, F, n    k, G, n    n, H    k, M, n    ph, k, n   
k, N    .+ , k, n    S, k, n    k, Z
Allowed substitution hints:    H( k)    N( n)    Z( n)

Proof of Theorem seqcoll2
StepHypRef Expression
1 seqcoll2.1b . . 3  |-  ( (
ph  /\  k  e.  S )  ->  (
k  .+  Z )  =  k )
2 fzssuz 11749 . . . 4  |-  ( M ... N )  C_  ( ZZ>= `  M )
3 seqcoll2.5 . . . . 5  |-  ( ph  ->  A  C_  ( M ... N ) )
4 seqcoll2.2 . . . . . . . 8  |-  ( ph  ->  G  Isom  <  ,  <  ( ( 1 ... ( # `
 A ) ) ,  A ) )
5 isof1o 6222 . . . . . . . 8  |-  ( G 
Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
64, 5syl 16 . . . . . . 7  |-  ( ph  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
7 f1of 5822 . . . . . . 7  |-  ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  G :
( 1 ... ( # `
 A ) ) --> A )
86, 7syl 16 . . . . . 6  |-  ( ph  ->  G : ( 1 ... ( # `  A
) ) --> A )
9 seqcoll2.3 . . . . . . . . . 10  |-  ( ph  ->  A  =/=  (/) )
10 fzfi 12084 . . . . . . . . . . . . 13  |-  ( M ... N )  e. 
Fin
11 ssfi 7759 . . . . . . . . . . . . 13  |-  ( ( ( M ... N
)  e.  Fin  /\  A  C_  ( M ... N ) )  ->  A  e.  Fin )
1210, 3, 11sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  Fin )
13 hasheq0 12435 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
1412, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  =  0  <->  A  =  (/) ) )
1514necon3bbid 2704 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ( # `  A )  =  0  <-> 
A  =/=  (/) ) )
169, 15mpbird 232 . . . . . . . . 9  |-  ( ph  ->  -.  ( # `  A
)  =  0 )
17 hashcl 12430 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
1812, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
19 elnn0 10818 . . . . . . . . . . 11  |-  ( (
# `  A )  e.  NN0  <->  ( ( # `  A )  e.  NN  \/  ( # `  A
)  =  0 ) )
2018, 19sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  A
)  e.  NN  \/  ( # `  A )  =  0 ) )
2120ord 377 . . . . . . . . 9  |-  ( ph  ->  ( -.  ( # `  A )  e.  NN  ->  ( # `  A
)  =  0 ) )
2216, 21mt3d 125 . . . . . . . 8  |-  ( ph  ->  ( # `  A
)  e.  NN )
23 nnuz 11141 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2555 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
1 ) )
25 eluzfz2 11719 . . . . . . 7  |-  ( (
# `  A )  e.  ( ZZ>= `  1 )  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
2624, 25syl 16 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
278, 26ffvelrnd 6033 . . . . 5  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  A )
283, 27sseldd 3500 . . . 4  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( M ... N ) )
292, 28sseldi 3497 . . 3  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( ZZ>= `  M
) )
30 elfzuz3 11710 . . . 4  |-  ( ( G `  ( # `  A ) )  e.  ( M ... N
)  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) ) )
3128, 30syl 16 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A ) ) ) )
32 fzss2 11748 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) )  -> 
( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3331, 32syl 16 . . . . . 6  |-  ( ph  ->  ( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3433sselda 3499 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... ( G `
 ( # `  A
) ) ) )  ->  k  e.  ( M ... N ) )
35 seqcoll2.6 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  S
)
3634, 35syldan 470 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( G `
 ( # `  A
) ) ) )  ->  ( F `  k )  e.  S
)
37 seqcoll2.c . . . 4  |-  ( (
ph  /\  ( k  e.  S  /\  n  e.  S ) )  -> 
( k  .+  n
)  e.  S )
3829, 36, 37seqcl 12129 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  e.  S
)
39 peano2uz 11159 . . . . . . . 8  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( ( G `  ( # `  A ) )  +  1 )  e.  ( ZZ>= `  M
) )
4029, 39syl 16 . . . . . . 7  |-  ( ph  ->  ( ( G `  ( # `  A ) )  +  1 )  e.  ( ZZ>= `  M
) )
41 fzss1 11747 . . . . . . 7  |-  ( ( ( G `  ( # `
 A ) )  +  1 )  e.  ( ZZ>= `  M )  ->  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) 
C_  ( M ... N ) )
4240, 41syl 16 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) 
C_  ( M ... N ) )
4342sselda 3499 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( M ... N ) )
44 eluzelre 11116 . . . . . . . . 9  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( G `  ( # `
 A ) )  e.  RR )
4529, 44syl 16 . . . . . . . 8  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  RR )
4645adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  e.  RR )
47 peano2re 9770 . . . . . . . 8  |-  ( ( G `  ( # `  A ) )  e.  RR  ->  ( ( G `  ( # `  A
) )  +  1 )  e.  RR )
4846, 47syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  e.  RR )
49 elfzelz 11713 . . . . . . . . 9  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  ZZ )
5049zred 10990 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  RR )
5150adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  RR )
5246ltp1d 10496 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  (
( G `  ( # `
 A ) )  +  1 ) )
53 elfzle1 11714 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5453adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5546, 48, 51, 52, 54ltletrd 9759 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  k
)
566adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
57 f1ocnv 5834 . . . . . . . . . . . . 13  |-  ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  `' G : A -1-1-onto-> ( 1 ... ( # `
 A ) ) )
5856, 57syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  `' G : A -1-1-onto-> ( 1 ... ( # `
 A ) ) )
59 f1of 5822 . . . . . . . . . . . 12  |-  ( `' G : A -1-1-onto-> ( 1 ... ( # `  A
) )  ->  `' G : A --> ( 1 ... ( # `  A
) ) )
6058, 59syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  `' G : A --> ( 1 ... ( # `  A
) ) )
61 simprr 757 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  k  e.  A )
6260, 61ffvelrnd 6033 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) )
63 elfzle2 11715 . . . . . . . . . 10  |-  ( ( `' G `  k )  e.  ( 1 ... ( # `  A
) )  ->  ( `' G `  k )  <_  ( # `  A
) )
6462, 63syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  <_  ( # `  A
) )
65 elfzelz 11713 . . . . . . . . . . . 12  |-  ( ( `' G `  k )  e.  ( 1 ... ( # `  A
) )  ->  ( `' G `  k )  e.  ZZ )
6662, 65syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ZZ )
6766zred 10990 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  RR )
6818adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e. 
NN0 )
6968nn0red 10874 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  RR )
7067, 69lenltd 9748 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( `' G `  k )  <_  ( # `
 A )  <->  -.  ( # `
 A )  < 
( `' G `  k ) ) )
7164, 70mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( # `  A )  <  ( `' G `  k ) )
724adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) )
7326adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  ( 1 ... ( # `
 A ) ) )
74 isorel 6223 . . . . . . . . . 10  |-  ( ( G  Isom  <  ,  <  ( ( 1 ... ( # `
 A ) ) ,  A )  /\  ( ( # `  A
)  e.  ( 1 ... ( # `  A
) )  /\  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) ) )  ->  ( ( # `  A )  <  ( `' G `  k )  <-> 
( G `  ( # `
 A ) )  <  ( G `  ( `' G `  k ) ) ) )
7572, 73, 62, 74syl12anc 1226 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  ( G `  ( `' G `  k )
) ) )
76 f1ocnvfv2 6184 . . . . . . . . . . 11  |-  ( ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  /\  k  e.  A )  ->  ( G `  ( `' G `  k )
)  =  k )
7756, 61, 76syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( G `  ( `' G `  k )
)  =  k )
7877breq2d 4468 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( G `  ( # `
 A ) )  <  ( G `  ( `' G `  k ) )  <->  ( G `  ( # `  A ) )  <  k ) )
7975, 78bitrd 253 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  k
) )
8071, 79mtbid 300 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( G `  ( # `  A ) )  < 
k )
8180expr 615 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( k  e.  A  ->  -.  ( G `  ( # `  A
) )  <  k
) )
8255, 81mt2d 117 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  -.  k  e.  A )
8343, 82eldifd 3482 . . . 4  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( ( M ... N )  \  A
) )
84 seqcoll2.7 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M ... N )  \  A
) )  ->  ( F `  k )  =  Z )
8583, 84syldan 470 . . 3  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( F `  k )  =  Z )
861, 29, 31, 38, 85seqid2 12155 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq M (  .+  ,  F ) `  N
) )
87 seqcoll2.1 . . 3  |-  ( (
ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )
88 seqcoll2.a . . 3  |-  ( ph  ->  Z  e.  S )
893, 2syl6ss 3511 . . 3  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
9033ssdifd 3636 . . . . 5  |-  ( ph  ->  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A )  C_  ( ( M ... N )  \  A
) )
9190sselda 3499 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A ) )  ->  k  e.  ( ( M ... N
)  \  A )
)
9291, 84syldan 470 . . 3  |-  ( (
ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A ) )  ->  ( F `  k )  =  Z )
93 seqcoll2.8 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( # `
 A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
 n ) ) )
9487, 1, 37, 88, 4, 26, 89, 36, 92, 93seqcoll 12515 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq 1 (  .+  ,  H ) `  ( # `
 A ) ) )
9586, 94eqtr3d 2500 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq 1 ( 
.+  ,  H ) `
 ( # `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468    C_ wss 3471   (/)c0 3793   class class class wbr 4456   `'ccnv 5007   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    < clt 9645    <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697    seqcseq 12109   #chash 12407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12110  df-hash 12408
This theorem is referenced by:  isercolllem3  13500  gsumval3OLD  17034  gsumval3  17037
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