MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqcoll2 Structured version   Unicode version

Theorem seqcoll2 12217
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 11499 . . . 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 6016 . . . . . . . 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 5641 . . . . . . 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 11794 . . . . . . . . . . . . 13  |-  ( M ... N )  e. 
Fin
11 ssfi 7533 . . . . . . . . . . . . 13  |-  ( ( ( M ... N
)  e.  Fin  /\  A  C_  ( M ... N ) )  ->  A  e.  Fin )
1210, 3, 11sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  Fin )
13 hasheq0 12131 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
1412, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  =  0  <->  A  =  (/) ) )
1514necon3bbid 2642 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ( # `  A )  =  0  <-> 
A  =/=  (/) ) )
169, 15mpbird 232 . . . . . . . . 9  |-  ( ph  ->  -.  ( # `  A
)  =  0 )
17 hashcl 12126 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
1812, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
19 elnn0 10581 . . . . . . . . . . 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 10896 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2533 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
1 ) )
25 eluzfz2 11459 . . . . . . 7  |-  ( (
# `  A )  e.  ( ZZ>= `  1 )  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
2624, 25syl 16 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
278, 26ffvelrnd 5844 . . . . 5  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  A )
283, 27sseldd 3357 . . . 4  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( M ... N ) )
292, 28sseldi 3354 . . 3  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( ZZ>= `  M
) )
30 elfzuz3 11450 . . . 4  |-  ( ( G `  ( # `  A ) )  e.  ( M ... N
)  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) ) )
3128, 30syl 16 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A ) ) ) )
32 fzss2 11498 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) )  -> 
( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3331, 32syl 16 . . . . . 6  |-  ( ph  ->  ( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3433sselda 3356 . . . . 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 11826 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  e.  S
)
39 peano2uz 10908 . . . . . . . 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 11497 . . . . . . 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 3356 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( M ... N ) )
44 eluzelre 10871 . . . . . . . . 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 9542 . . . . . . . 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 11453 . . . . . . . . 9  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  ZZ )
5049zred 10747 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  RR )
5150adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  RR )
5246ltp1d 10263 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  (
( G `  ( # `
 A ) )  +  1 ) )
53 elfzle1 11454 . . . . . . . 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 9531 . . . . . 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 5653 . . . . . . . . . . . . 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 5641 . . . . . . . . . . . 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 756 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  k  e.  A )
6260, 61ffvelrnd 5844 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) )
63 elfzle2 11455 . . . . . . . . . 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 11453 . . . . . . . . . . . 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 10747 . . . . . . . . . 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 10637 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  RR )
7067, 69lenltd 9520 . . . . . . . . 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 6017 . . . . . . . . . 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 1216 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  ( G `  ( `' G `  k )
) ) )
76 f1ocnvfv2 5984 . . . . . . . . . . 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 4304 . . . . . . . . 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 3339 . . . 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 11852 . 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 3368 . . 3  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
9033ssdifd 3492 . . . . 5  |-  ( ph  ->  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A )  C_  ( ( M ... N )  \  A
) )
9190sselda 3356 . . . 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 12216 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq 1 (  .+  ,  H ) `  ( # `
 A ) ) )
9586, 94eqtr3d 2477 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 1369    e. wcel 1756    =/= wne 2606    \ cdif 3325    C_ wss 3328   (/)c0 3637   class class class wbr 4292   `'ccnv 4839   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418    Isom wiso 5419  (class class class)co 6091   Fincfn 7310   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419   NNcn 10322   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437    seqcseq 11806   #chash 12103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-seq 11807  df-hash 12104
This theorem is referenced by:  isercolllem3  13144  gsumval3OLD  16382  gsumval3  16385
  Copyright terms: Public domain W3C validator