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Theorem seqcoll2 11402
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 10832 . . . 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 5822 . . . . . . . 8  |-  ( G 
Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
64, 5syl 15 . . . . . . 7  |-  ( ph  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
7 f1of 5472 . . . . . . 7  |-  ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  G :
( 1 ... ( # `
 A ) ) --> A )
86, 7syl 15 . . . . . 6  |-  ( ph  ->  G : ( 1 ... ( # `  A
) ) --> A )
9 seqcoll2.3 . . . . . . . . . 10  |-  ( ph  ->  A  =/=  (/) )
10 fzfi 11034 . . . . . . . . . . . . 13  |-  ( M ... N )  e. 
Fin
11 ssfi 7083 . . . . . . . . . . . . 13  |-  ( ( ( M ... N
)  e.  Fin  /\  A  C_  ( M ... N ) )  ->  A  e.  Fin )
1210, 3, 11sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  Fin )
13 hasheq0 11353 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
1412, 13syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  A
)  =  0  <->  A  =  (/) ) )
1514necon3bbid 2480 . . . . . . . . . 10  |-  ( ph  ->  ( -.  ( # `  A )  =  0  <-> 
A  =/=  (/) ) )
169, 15mpbird 223 . . . . . . . . 9  |-  ( ph  ->  -.  ( # `  A
)  =  0 )
17 hashcl 11350 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
1812, 17syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
19 elnn0 9967 . . . . . . . . . . 11  |-  ( (
# `  A )  e.  NN0  <->  ( ( # `  A )  e.  NN  \/  ( # `  A
)  =  0 ) )
2018, 19sylib 188 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  A
)  e.  NN  \/  ( # `  A )  =  0 ) )
2120ord 366 . . . . . . . . 9  |-  ( ph  ->  ( -.  ( # `  A )  e.  NN  ->  ( # `  A
)  =  0 ) )
2216, 21mt3d 117 . . . . . . . 8  |-  ( ph  ->  ( # `  A
)  e.  NN )
23 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2373 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
1 ) )
25 eluzfz2 10804 . . . . . . 7  |-  ( (
# `  A )  e.  ( ZZ>= `  1 )  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
2624, 25syl 15 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ( 1 ... ( # `  A
) ) )
27 ffvelrn 5663 . . . . . 6  |-  ( ( G : ( 1 ... ( # `  A
) ) --> A  /\  ( # `  A )  e.  ( 1 ... ( # `  A
) ) )  -> 
( G `  ( # `
 A ) )  e.  A )
288, 26, 27syl2anc 642 . . . . 5  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  A )
293, 28sseldd 3181 . . . 4  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( M ... N ) )
302, 29sseldi 3178 . . 3  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  ( ZZ>= `  M
) )
31 elfzuz3 10795 . . . 4  |-  ( ( G `  ( # `  A ) )  e.  ( M ... N
)  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) ) )
3229, 31syl 15 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  ( G `  ( # `  A ) ) ) )
33 fzss2 10831 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( G `  ( # `  A
) ) )  -> 
( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3432, 33syl 15 . . . . . 6  |-  ( ph  ->  ( M ... ( G `  ( # `  A
) ) )  C_  ( M ... N ) )
3534sselda 3180 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... ( G `
 ( # `  A
) ) ) )  ->  k  e.  ( M ... N ) )
36 seqcoll2.6 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  S
)
3735, 36syldan 456 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( G `
 ( # `  A
) ) ) )  ->  ( F `  k )  e.  S
)
38 seqcoll2.c . . . 4  |-  ( (
ph  /\  ( k  e.  S  /\  n  e.  S ) )  -> 
( k  .+  n
)  e.  S )
3930, 37, 38seqcl 11066 . . 3  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  e.  S
)
40 peano2uz 10272 . . . . . . . 8  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( ( G `  ( # `  A ) )  +  1 )  e.  ( ZZ>= `  M
) )
4130, 40syl 15 . . . . . . 7  |-  ( ph  ->  ( ( G `  ( # `  A ) )  +  1 )  e.  ( ZZ>= `  M
) )
42 fzss1 10830 . . . . . . 7  |-  ( ( ( G `  ( # `
 A ) )  +  1 )  e.  ( ZZ>= `  M )  ->  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) 
C_  ( M ... N ) )
4341, 42syl 15 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) 
C_  ( M ... N ) )
4443sselda 3180 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( M ... N ) )
45 eluzelre 10239 . . . . . . . . 9  |-  ( ( G `  ( # `  A ) )  e.  ( ZZ>= `  M )  ->  ( G `  ( # `
 A ) )  e.  RR )
4630, 45syl 15 . . . . . . . 8  |-  ( ph  ->  ( G `  ( # `
 A ) )  e.  RR )
4746adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  e.  RR )
48 peano2re 8985 . . . . . . . 8  |-  ( ( G `  ( # `  A ) )  e.  RR  ->  ( ( G `  ( # `  A
) )  +  1 )  e.  RR )
4947, 48syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  e.  RR )
50 elfzelz 10798 . . . . . . . . 9  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  ZZ )
5150zred 10117 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  k  e.  RR )
5251adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  RR )
5347ltp1d 9687 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  (
( G `  ( # `
 A ) )  +  1 ) )
54 elfzle1 10799 . . . . . . . 8  |-  ( k  e.  ( ( ( G `  ( # `  A ) )  +  1 ) ... N
)  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5554adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( ( G `  ( # `  A
) )  +  1 )  <_  k )
5647, 49, 52, 53, 55ltletrd 8976 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( G `  ( # `  A
) )  <  k
)
576adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
58 f1ocnv 5485 . . . . . . . . . . . . 13  |-  ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  `' G : A -1-1-onto-> ( 1 ... ( # `
 A ) ) )
5957, 58syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  `' G : A -1-1-onto-> ( 1 ... ( # `
 A ) ) )
60 f1of 5472 . . . . . . . . . . . 12  |-  ( `' G : A -1-1-onto-> ( 1 ... ( # `  A
) )  ->  `' G : A --> ( 1 ... ( # `  A
) ) )
6159, 60syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  `' G : A --> ( 1 ... ( # `  A
) ) )
62 simprr 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  k  e.  A )
63 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( `' G : A --> ( 1 ... ( # `  A
) )  /\  k  e.  A )  ->  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) )
6461, 62, 63syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) )
65 elfzle2 10800 . . . . . . . . . 10  |-  ( ( `' G `  k )  e.  ( 1 ... ( # `  A
) )  ->  ( `' G `  k )  <_  ( # `  A
) )
6664, 65syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  <_  ( # `  A
) )
67 elfzelz 10798 . . . . . . . . . . . 12  |-  ( ( `' G `  k )  e.  ( 1 ... ( # `  A
) )  ->  ( `' G `  k )  e.  ZZ )
6864, 67syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  ZZ )
6968zred 10117 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( `' G `  k )  e.  RR )
7018adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e. 
NN0 )
7170nn0red 10019 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  RR )
7269, 71lenltd 8965 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( `' G `  k )  <_  ( # `
 A )  <->  -.  ( # `
 A )  < 
( `' G `  k ) ) )
7366, 72mpbid 201 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( # `  A )  <  ( `' G `  k ) )
744adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  G  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) )
7526adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( # `
 A )  e.  ( 1 ... ( # `
 A ) ) )
76 isorel 5823 . . . . . . . . . 10  |-  ( ( G  Isom  <  ,  <  ( ( 1 ... ( # `
 A ) ) ,  A )  /\  ( ( # `  A
)  e.  ( 1 ... ( # `  A
) )  /\  ( `' G `  k )  e.  ( 1 ... ( # `  A
) ) ) )  ->  ( ( # `  A )  <  ( `' G `  k )  <-> 
( G `  ( # `
 A ) )  <  ( G `  ( `' G `  k ) ) ) )
7774, 75, 64, 76syl12anc 1180 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  ( G `  ( `' G `  k )
) ) )
78 f1ocnvfv2 5793 . . . . . . . . . . 11  |-  ( ( G : ( 1 ... ( # `  A
) ) -1-1-onto-> A  /\  k  e.  A )  ->  ( G `  ( `' G `  k )
)  =  k )
7957, 62, 78syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  ( G `  ( `' G `  k )
)  =  k )
8079breq2d 4035 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( G `  ( # `
 A ) )  <  ( G `  ( `' G `  k ) )  <->  ( G `  ( # `  A ) )  <  k ) )
8177, 80bitrd 244 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  (
( # `  A )  <  ( `' G `  k )  <->  ( G `  ( # `  A
) )  <  k
) )
8273, 81mtbid 291 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N )  /\  k  e.  A
) )  ->  -.  ( G `  ( # `  A ) )  < 
k )
8382expr 598 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( k  e.  A  ->  -.  ( G `  ( # `  A
) )  <  k
) )
8456, 83mt2d 109 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  -.  k  e.  A )
85 eldif 3162 . . . . 5  |-  ( k  e.  ( ( M ... N )  \  A )  <->  ( k  e.  ( M ... N
)  /\  -.  k  e.  A ) )
8644, 84, 85sylanbrc 645 . . . 4  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  k  e.  ( ( M ... N )  \  A
) )
87 seqcoll2.7 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M ... N )  \  A
) )  ->  ( F `  k )  =  Z )
8886, 87syldan 456 . . 3  |-  ( (
ph  /\  k  e.  ( ( ( G `
 ( # `  A
) )  +  1 ) ... N ) )  ->  ( F `  k )  =  Z )
891, 30, 32, 39, 88seqid2 11092 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq  M (  .+  ,  F ) `  N
) )
90 seqcoll2.1 . . 3  |-  ( (
ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )
91 seqcoll2.a . . 3  |-  ( ph  ->  Z  e.  S )
923, 2syl6ss 3191 . . 3  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
93 ssdif 3311 . . . . . 6  |-  ( ( M ... ( G `
 ( # `  A
) ) )  C_  ( M ... N )  ->  ( ( M ... ( G `  ( # `  A ) ) )  \  A
)  C_  ( ( M ... N )  \  A ) )
9434, 93syl 15 . . . . 5  |-  ( ph  ->  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A )  C_  ( ( M ... N )  \  A
) )
9594sselda 3180 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A ) )  ->  k  e.  ( ( M ... N
)  \  A )
)
9695, 87syldan 456 . . 3  |-  ( (
ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
\  A ) )  ->  ( F `  k )  =  Z )
97 seqcoll2.8 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( # `
 A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
 n ) ) )
9890, 1, 38, 91, 4, 26, 92, 37, 96, 97seqcoll 11401 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 ( G `  ( # `  A ) ) )  =  (  seq  1 (  .+  ,  H ) `  ( # `
 A ) ) )
9989, 98eqtr3d 2317 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  1 ( 
.+  ,  H ) `
 ( # `  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   (/)c0 3455   class class class wbr 4023   `'ccnv 4688   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   Fincfn 6863   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   #chash 11337
This theorem is referenced by:  isercolllem3  12140  gsumval3  15191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-hash 11338
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