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Theorem seqcoll2 12628
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 11839 . . . 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 6216 . . . . . . . 8  |-  ( G 
Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
64, 5syl 17 . . . . . . 7  |-  ( ph  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
7 f1of 5814 . . . . . . 7  |-  ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  G :
( 1 ... ( # `
 A ) ) --> A )
86, 7syl 17 . . . . . 6  |-  ( ph  ->  G : ( 1 ... ( # `  A
) ) --> A )
9 seqcoll2.3 . . . . . . . . . 10  |-  ( ph  ->  A  =/=  (/) )
10 fzfi 12185 . . . . . . . . . . . . 13  |-  ( M ... N )  e. 
Fin
11 ssfi 7792 . . . . . . . . . . . . 13  |-  ( ( ( M ... N
)  e.  Fin  /\  A  C_  ( M ... N ) )  ->  A  e.  Fin )
1210, 3, 11sylancr 669 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  Fin )
13 hasheq0 12544 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
1412, 13syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  =  0  <->  A  =  (/) ) )
1514necon3bbid 2661 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ( # `  A )  =  0  <-> 
A  =/=  (/) ) )
169, 15mpbird 236 . . . . . . . . 9  |-  ( ph  ->  -.  ( # `  A
)  =  0 )
17 hashcl 12538 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
1812, 17syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
19 elnn0 10871 . . . . . . . . . . 11  |-  ( (
# `  A )  e.  NN0  <->  ( ( # `  A )  e.  NN  \/  ( # `  A
)  =  0 ) )
2018, 19sylib 200 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  A
)  e.  NN  \/  ( # `  A )  =  0 ) )
2120ord 379 . . . . . . . . 9  |-  ( ph  ->  ( -.  ( # `  A )  e.  NN  ->  ( # `  A
)  =  0 ) )
2216, 21mt3d 129 . . . . . . . 8  |-  ( ph  ->  ( # `  A
)  e.  NN )
23 nnuz 11194 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2539 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
1 ) )
25 eluzfz2 11807 . . . . . . 7  |-  ( (
# `  A )  e.  ( ZZ>= `  1 )  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
2624, 25syl 17 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
278, 26ffvelrnd 6023 . . . . 5  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  A )
283, 27sseldd 3433 . . . 4  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( M ... N ) )
292, 28sseldi 3430 . . 3  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( ZZ>= `  M
) )
30 elfzuz3 11797 . . . 4  |-  ( ( G `  ( # `  A ) )  e.  ( M ... N
)  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) ) )
3128, 30syl 17 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A ) ) ) )
32 fzss2 11838 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) )  -> 
( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3331, 32syl 17 . . . . . 6  |-  ( ph  ->  ( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3433sselda 3432 . . . . 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 473 . . . 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 12233 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  e.  S
)
39 peano2uz 11212 . . . . . . . 8  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( ( G `  ( # `  A ) )  +  1 )  e.  ( ZZ>= `  M
) )
4029, 39syl 17 . . . . . . 7  |-  ( ph  ->  ( ( G `  ( # `  A ) )  +  1 )  e.  ( ZZ>= `  M
) )
41 fzss1 11837 . . . . . . 7  |-  ( ( ( G `  ( # `
 A ) )  +  1 )  e.  ( ZZ>= `  M )  ->  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) 
C_  ( M ... N ) )
4240, 41syl 17 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) 
C_  ( M ... N ) )
4342sselda 3432 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( M ... N ) )
44 eluzelre 11169 . . . . . . . . 9  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( G `  ( # `
 A ) )  e.  RR )
4529, 44syl 17 . . . . . . . 8  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  RR )
4645adantr 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  e.  RR )
47 peano2re 9806 . . . . . . . 8  |-  ( ( G `  ( # `  A ) )  e.  RR  ->  ( ( G `  ( # `  A
) )  +  1 )  e.  RR )
4846, 47syl 17 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  e.  RR )
49 elfzelz 11800 . . . . . . . . 9  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  ZZ )
5049zred 11040 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  RR )
5150adantl 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  RR )
5246ltp1d 10537 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  (
( G `  ( # `
 A ) )  +  1 ) )
53 elfzle1 11802 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5453adantl 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5546, 48, 51, 52, 54ltletrd 9795 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  k
)
566adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
57 f1ocnv 5826 . . . . . . . . . . . . 13  |-  ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  `' G : A -1-1-onto-> ( 1 ... ( # `
 A ) ) )
5856, 57syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  `' G : A -1-1-onto-> ( 1 ... ( # `
 A ) ) )
59 f1of 5814 . . . . . . . . . . . 12  |-  ( `' G : A -1-1-onto-> ( 1 ... ( # `  A
) )  ->  `' G : A --> ( 1 ... ( # `  A
) ) )
6058, 59syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  `' G : A --> ( 1 ... ( # `  A
) ) )
61 simprr 766 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  k  e.  A )
6260, 61ffvelrnd 6023 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) )
63 elfzle2 11803 . . . . . . . . . 10  |-  ( ( `' G `  k )  e.  ( 1 ... ( # `  A
) )  ->  ( `' G `  k )  <_  ( # `  A
) )
6462, 63syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  <_  ( # `  A
) )
65 elfzelz 11800 . . . . . . . . . . . 12  |-  ( ( `' G `  k )  e.  ( 1 ... ( # `  A
) )  ->  ( `' G `  k )  e.  ZZ )
6662, 65syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ZZ )
6766zred 11040 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  RR )
6818adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e. 
NN0 )
6968nn0red 10926 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  RR )
7067, 69lenltd 9781 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( `' G `  k )  <_  ( # `
 A )  <->  -.  ( # `
 A )  < 
( `' G `  k ) ) )
7164, 70mpbid 214 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( # `  A )  <  ( `' G `  k ) )
724adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) )
7326adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  ( 1 ... ( # `
 A ) ) )
74 isorel 6217 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  ( G `  ( `' G `  k )
) ) )
76 f1ocnvfv2 6176 . . . . . . . . . . 11  |-  ( ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  /\  k  e.  A )  ->  ( G `  ( `' G `  k )
)  =  k )
7756, 61, 76syl2anc 667 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( G `  ( `' G `  k )
)  =  k )
7877breq2d 4414 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( G `  ( # `
 A ) )  <  ( G `  ( `' G `  k ) )  <->  ( G `  ( # `  A ) )  <  k ) )
7975, 78bitrd 257 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  k
) )
8071, 79mtbid 302 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( G `  ( # `  A ) )  < 
k )
8180expr 620 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( k  e.  A  ->  -.  ( G `  ( # `  A
) )  <  k
) )
8255, 81mt2d 121 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  -.  k  e.  A )
8343, 82eldifd 3415 . . . 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 473 . . 3  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( F `  k )  =  Z )
861, 29, 31, 38, 85seqid2 12259 . 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 3444 . . 3  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
9033ssdifd 3569 . . . . 5  |-  ( ph  ->  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A )  C_  ( ( M ... N )  \  A
) )
9190sselda 3432 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A ) )  ->  k  e.  ( ( M ... N
)  \  A )
)
9291, 84syldan 473 . . 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 12627 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq 1 (  .+  ,  H ) `  ( # `
 A ) ) )
9586, 94eqtr3d 2487 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq 1 ( 
.+  ,  H ) `
 ( # `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622    \ cdif 3401    C_ wss 3404   (/)c0 3731   class class class wbr 4402   `'ccnv 4833   -->wf 5578   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583  (class class class)co 6290   Fincfn 7569   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784    seqcseq 12213   #chash 12515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-seq 12214  df-hash 12516
This theorem is referenced by:  isercolllem3  13730  gsumval3  17541
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