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Theorem seqcoll2 11713
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 11093 . . . 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 6045 . . . . . . . 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 5674 . . . . . . 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 11311 . . . . . . . . . . . . 13  |-  ( M ... N )  e. 
Fin
11 ssfi 7329 . . . . . . . . . . . . 13  |-  ( ( ( M ... N
)  e.  Fin  /\  A  C_  ( M ... N ) )  ->  A  e.  Fin )
1210, 3, 11sylancr 645 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  Fin )
13 hasheq0 11644 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
1412, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  =  0  <->  A  =  (/) ) )
1514necon3bbid 2635 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ( # `  A )  =  0  <-> 
A  =/=  (/) ) )
169, 15mpbird 224 . . . . . . . . 9  |-  ( ph  ->  -.  ( # `  A
)  =  0 )
17 hashcl 11639 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
1812, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
19 elnn0 10223 . . . . . . . . . . 11  |-  ( (
# `  A )  e.  NN0  <->  ( ( # `  A )  e.  NN  \/  ( # `  A
)  =  0 ) )
2018, 19sylib 189 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  A
)  e.  NN  \/  ( # `  A )  =  0 ) )
2120ord 367 . . . . . . . . 9  |-  ( ph  ->  ( -.  ( # `  A )  e.  NN  ->  ( # `  A
)  =  0 ) )
2216, 21mt3d 119 . . . . . . . 8  |-  ( ph  ->  ( # `  A
)  e.  NN )
23 nnuz 10521 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2526 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
1 ) )
25 eluzfz2 11065 . . . . . . 7  |-  ( (
# `  A )  e.  ( ZZ>= `  1 )  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
2624, 25syl 16 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
278, 26ffvelrnd 5871 . . . . 5  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  A )
283, 27sseldd 3349 . . . 4  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( M ... N ) )
292, 28sseldi 3346 . . 3  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( ZZ>= `  M
) )
30 elfzuz3 11056 . . . 4  |-  ( ( G `  ( # `  A ) )  e.  ( M ... N
)  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) ) )
3128, 30syl 16 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A ) ) ) )
32 fzss2 11092 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) )  -> 
( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3331, 32syl 16 . . . . . 6  |-  ( ph  ->  ( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3433sselda 3348 . . . . 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 457 . . . 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 11343 . . 3  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  e.  S
)
39 peano2uz 10530 . . . . . . . 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 11091 . . . . . . 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 3348 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( M ... N ) )
44 eluzelre 10497 . . . . . . . . 9  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( G `  ( # `
 A ) )  e.  RR )
4529, 44syl 16 . . . . . . . 8  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  RR )
4645adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  e.  RR )
47 peano2re 9239 . . . . . . . 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 11059 . . . . . . . . 9  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  ZZ )
5049zred 10375 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  RR )
5150adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  RR )
5246ltp1d 9941 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  (
( G `  ( # `
 A ) )  +  1 ) )
53 elfzle1 11060 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5453adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5546, 48, 51, 52, 54ltletrd 9230 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  k
)
566adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
57 f1ocnv 5687 . . . . . . . . . . . . 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 5674 . . . . . . . . . . . 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 734 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  k  e.  A )
6260, 61ffvelrnd 5871 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) )
63 elfzle2 11061 . . . . . . . . . 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 11059 . . . . . . . . . . . 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 10375 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  RR )
6818adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e. 
NN0 )
6968nn0red 10275 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  RR )
7067, 69lenltd 9219 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( `' G `  k )  <_  ( # `
 A )  <->  -.  ( # `
 A )  < 
( `' G `  k ) ) )
7164, 70mpbid 202 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( # `  A )  <  ( `' G `  k ) )
724adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) )
7326adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  ( 1 ... ( # `
 A ) ) )
74 isorel 6046 . . . . . . . . . 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 1182 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  ( G `  ( `' G `  k )
) ) )
76 f1ocnvfv2 6015 . . . . . . . . . . 11  |-  ( ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  /\  k  e.  A )  ->  ( G `  ( `' G `  k )
)  =  k )
7756, 61, 76syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( G `  ( `' G `  k )
)  =  k )
7877breq2d 4224 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( G `  ( # `
 A ) )  <  ( G `  ( `' G `  k ) )  <->  ( G `  ( # `  A ) )  <  k ) )
7975, 78bitrd 245 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  k
) )
8071, 79mtbid 292 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( G `  ( # `  A ) )  < 
k )
8180expr 599 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( k  e.  A  ->  -.  ( G `  ( # `  A
) )  <  k
) )
8255, 81mt2d 111 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  -.  k  e.  A )
8343, 82eldifd 3331 . . . 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 457 . . 3  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( F `  k )  =  Z )
861, 29, 31, 38, 85seqid2 11369 . 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 3360 . . 3  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
9033ssdifd 3483 . . . . 5  |-  ( ph  ->  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A )  C_  ( ( M ... N )  \  A
) )
9190sselda 3348 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A ) )  ->  k  e.  ( ( M ... N
)  \  A )
)
9291, 84syldan 457 . . 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 11712 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq  1 (  .+  ,  H ) `  ( # `
 A ) ) )
9586, 94eqtr3d 2470 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  1 ( 
.+  ,  H ) `
 ( # `  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    C_ wss 3320   (/)c0 3628   class class class wbr 4212   `'ccnv 4877   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455  (class class class)co 6081   Fincfn 7109   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121   NNcn 10000   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323   #chash 11618
This theorem is referenced by:  isercolllem3  12460  gsumval3  15514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-hash 11619
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