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Theorem seqid2 11848
Description: The last few terms of a sequence that ends with all zeroes (or whatever the identity  Z is for operation  .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqid2.1  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
seqid2.2  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seqid2.3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
seqid2.4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  e.  S )
seqid2.5  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
Assertion
Ref Expression
seqid2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  (  seq M ( 
.+  ,  F ) `
 N ) )
Distinct variable groups:    x, F    x, K    x, M    x, N    ph, x    x, S    x, 
.+    x, Z

Proof of Theorem seqid2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 seqid2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 11455 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2501 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5688 . . . . . . 7  |-  ( x  =  K  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  K
) )
65eqeq2d 2452 . . . . . 6  |-  ( x  =  K  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 K ) ) )
74, 6imbi12d 320 . . . . 5  |-  ( x  =  K  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 x ) )  <-> 
( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 K ) ) ) )
87imbi2d 316 . . . 4  |-  ( x  =  K  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 K ) ) ) ) )
9 eleq1 2501 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5688 . . . . . . 7  |-  ( x  =  n  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  n
) )
1110eqeq2d 2452 . . . . . 6  |-  ( x  =  n  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 n ) ) )
129, 11imbi12d 320 . . . . 5  |-  ( x  =  n  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 x ) )  <-> 
( n  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 n ) ) ) )
1312imbi2d 316 . . . 4  |-  ( x  =  n  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( n  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 n ) ) ) ) )
14 eleq1 2501 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5688 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2452 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
1714, 16imbi12d 320 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 x ) )  <-> 
( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) )
1817imbi2d 316 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) ) )
19 eleq1 2501 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5688 . . . . . . 7  |-  ( x  =  N  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  N
) )
2120eqeq2d 2452 . . . . . 6  |-  ( x  =  N  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )
2219, 21imbi12d 320 . . . . 5  |-  ( x  =  N  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 x ) )  <-> 
( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) ) )
2322imbi2d 316 . . . 4  |-  ( x  =  N  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) ) ) )
24 eqidd 2442 . . . . 5  |-  ( K  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
2524a1ii 27 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) ) ) )
26 peano2fzr 11459 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 463 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 612 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 75 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
) ) ) )
30 oveq1 6097 . . . . . . . . . 10  |-  ( (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
)  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) )
31 eluzp1p1 10882 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3231ad2antrl 722 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
33 elfzuz3 11446 . . . . . . . . . . . . . . . 16  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3433ad2antll 723 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
35 elfzuzb 11443 . . . . . . . . . . . . . . 15  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
3632, 34, 35sylanbrc 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
37 seqid2.5 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
3837ralrimiva 2797 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3938adantr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
40 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( n  + 
1 )  ->  ( F `  x )  =  ( F `  ( n  +  1
) ) )
4140eqeq1d 2449 . . . . . . . . . . . . . . 15  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
4241rspcv 3066 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) ( F `  x
)  =  Z  -> 
( F `  (
n  +  1 ) )  =  Z ) )
4336, 39, 42sylc 60 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4443oveq2d 6106 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M
(  .+  ,  F
) `  K )  .+  Z ) )
45 seqid2.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  e.  S )
46 seqid2.1 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
4746ralrimiva 2797 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 oveq1 6097 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq M
(  .+  ,  F
) `  K )  ->  ( x  .+  Z
)  =  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )
)
49 id 22 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq M
(  .+  ,  F
) `  K )  ->  x  =  (  seq M (  .+  ,  F ) `  K
) )
5048, 49eqeq12d 2455 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq M
(  .+  ,  F
) `  K )  ->  ( ( x  .+  Z )  =  x  <-> 
( (  seq M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq M (  .+  ,  F ) `  K
) ) )
5150rspcv 3066 . . . . . . . . . . . . . 14  |-  ( (  seq M (  .+  ,  F ) `  K
)  e.  S  -> 
( A. x  e.  S  ( x  .+  Z )  =  x  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq M ( 
.+  ,  F ) `
 K ) ) )
5245, 47, 51sylc 60 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq M (  .+  ,  F ) `  K
) )
5352adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq M ( 
.+  ,  F ) `
 K ) )
5444, 53eqtr2d 2474 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
55 simprl 750 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  K )
)
56 seqid2.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
5756adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
58 uztrn 10873 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5955, 57, 58syl2anc 656 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
60 seqp1 11817 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6159, 60syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6254, 61eqeq12d 2455 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  <->  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) ) )
6330, 62syl5ibr 221 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) ) )
6463expr 612 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) )
6564a2d 26 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) )
6629, 65syld 44 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) )
6766expcom 435 . . . . 5  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) ) )
6867a2d 26 . . . 4  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
) ) )  -> 
( ph  ->  ( ( n  +  1 )  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) ) )
698, 13, 18, 23, 25, 68uzind4 10908 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) ) )
701, 69mpcom 36 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )
713, 70mpd 15 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  (  seq M ( 
.+  ,  F ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   ` cfv 5415  (class class class)co 6090   1c1 9279    + caddc 9281   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803
This theorem is referenced by:  seqcoll  12212  seqcoll2  12213  fsumcvg  13185  ovolicc1  20899  lgsdilem2  22613  fprodcvg  27356
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