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Theorem iseqid2 9616
Description: The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for  .+) are all the same. (Contributed by Jim Kingdon, 5-Mar-2022.)
Hypotheses
Ref Expression
iseqid2.1  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
iseqid2.2  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
iseqid2.3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
iseqid2.4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  e.  S )
iseqid2.5  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
iseqid2.s  |-  ( ph  ->  S  e.  V )
iseqid2.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqid2.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
iseqid2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  N ) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, K, y    x, M, y   
x, N, y    x, S, y    x, Z    ph, x, y
Allowed substitution hints:    V( x, y)    Z( y)

Proof of Theorem iseqid2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 iseqid2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 9179 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2145 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5229 . . . . . . 7  |-  ( x  =  K  ->  (  seq M (  .+  ,  F ,  S ) `  x )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) )
65eqeq2d 2094 . . . . . 6  |-  ( x  =  K  ->  (
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  x
)  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  K ) ) )
74, 6imbi12d 232 . . . . 5  |-  ( x  =  K  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  x ) )  <->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) ) ) )
87imbi2d 228 . . . 4  |-  ( x  =  K  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 x ) ) )  <->  ( ph  ->  ( K  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  K
) ) ) ) )
9 eleq1 2145 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5229 . . . . . . 7  |-  ( x  =  n  ->  (  seq M (  .+  ,  F ,  S ) `  x )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) )
1110eqeq2d 2094 . . . . . 6  |-  ( x  =  n  ->  (
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  x
)  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  n ) ) )
129, 11imbi12d 232 . . . . 5  |-  ( x  =  n  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  x ) )  <->  ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) ) ) )
1312imbi2d 228 . . . 4  |-  ( x  =  n  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 x ) ) )  <->  ( ph  ->  ( n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
) ) ) ) )
14 eleq1 2145 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5229 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  .+  ,  F ,  S ) `  x )  =  (  seq M (  .+  ,  F ,  S ) `
 ( n  + 
1 ) ) )
1615eqeq2d 2094 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  x
)  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  ( n  +  1 ) ) ) )
1714, 16imbi12d 232 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  x ) )  <->  ( (
n  +  1 )  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 ( n  + 
1 ) ) ) ) )
1817imbi2d 228 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 x ) ) )  <->  ( ph  ->  ( ( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  (
n  +  1 ) ) ) ) ) )
19 eleq1 2145 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5229 . . . . . . 7  |-  ( x  =  N  ->  (  seq M (  .+  ,  F ,  S ) `  x )  =  (  seq M (  .+  ,  F ,  S ) `
 N ) )
2120eqeq2d 2094 . . . . . 6  |-  ( x  =  N  ->  (
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  x
)  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  N ) ) )
2219, 21imbi12d 232 . . . . 5  |-  ( x  =  N  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  x ) )  <->  ( N  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 N ) ) ) )
2322imbi2d 228 . . . 4  |-  ( x  =  N  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 x ) ) )  <->  ( ph  ->  ( N  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  N
) ) ) ) )
24 eqidd 2084 . . . . 5  |-  ( K  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) )
25242a1i 27 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) ) ) )
26 peano2fzr 9184 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 367 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 74 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
) ) ) )
30 oveq1 5570 . . . . . . . . . 10  |-  ( (  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  K )  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  S ) `
 n )  .+  ( F `  ( n  +  1 ) ) ) )
31 fveq2 5229 . . . . . . . . . . . . . . 15  |-  ( x  =  ( n  + 
1 )  ->  ( F `  x )  =  ( F `  ( n  +  1
) ) )
3231eqeq1d 2091 . . . . . . . . . . . . . 14  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
33 iseqid2.5 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
3433ralrimiva 2439 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3534adantr 270 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
36 eluzp1p1 8777 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3736ad2antrl 474 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
38 elfzuz3 9170 . . . . . . . . . . . . . . . 16  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3938ad2antll 475 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
40 elfzuzb 9167 . . . . . . . . . . . . . . 15  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
4137, 39, 40sylanbrc 408 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
4232, 35, 41rspcdva 2715 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4342oveq2d 5579 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  K )  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  S ) `
 K )  .+  Z ) )
44 oveq1 5570 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  K
)  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  K
)  .+  Z )
)
45 id 19 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  K
)  ->  x  =  (  seq M (  .+  ,  F ,  S ) `
 K ) )
4644, 45eqeq12d 2097 . . . . . . . . . . . . . 14  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  K
)  ->  ( (
x  .+  Z )  =  x  <->  ( (  seq M (  .+  ,  F ,  S ) `  K )  .+  Z
)  =  (  seq M (  .+  ,  F ,  S ) `  K ) ) )
47 iseqid2.1 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
4847ralrimiva 2439 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
49 iseqid2.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  e.  S )
5046, 48, 49rspcdva 2715 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (  seq M
(  .+  ,  F ,  S ) `  K
)  .+  Z )  =  (  seq M ( 
.+  ,  F ,  S ) `  K
) )
5150adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  K )  .+  Z
)  =  (  seq M (  .+  ,  F ,  S ) `  K ) )
5243, 51eqtr2d 2116 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
53 simprl 498 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  K )
)
54 iseqid2.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
5554adantr 270 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
56 uztrn 8768 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5753, 55, 56syl2anc 403 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
58 iseqid2.f . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
5958adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  M ) )  ->  ( F `  x )  e.  S
)
60 iseqid2.cl . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6160adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6257, 59, 61iseqp1 9590 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ,  S ) `  ( n  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6352, 62eqeq12d 2097 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 ( n  + 
1 ) )  <->  ( (  seq M (  .+  ,  F ,  S ) `  K )  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  S ) `
 n )  .+  ( F `  ( n  +  1 ) ) ) ) )
6430, 63syl5ibr 154 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 n )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  (
n  +  1 ) ) ) )
6564expr 367 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 ( n  + 
1 ) ) ) ) )
6665a2d 26 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  (
n  +  1 ) ) ) ) )
6729, 66syld 44 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq M ( 
.+  ,  F ,  S ) `  (
n  +  1 ) ) ) ) )
6867expcom 114 . . . . 5  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) )  ->  ( ( n  +  1 )  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 ( n  + 
1 ) ) ) ) ) )
6968a2d 26 . . . 4  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) ) )  ->  ( ph  ->  ( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  ( n  +  1 ) ) ) ) ) )
708, 13, 18, 23, 25, 69uzind4 8809 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  N ) ) ) )
711, 70mpcom 36 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  N ) ) )
723, 71mpd 13 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  =  (  seq M (  .+  ,  F ,  S ) `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2353   ` cfv 4952  (class class class)co 5563   1c1 7096    + caddc 7098   ZZcz 8484   ZZ>=cuz 8752   ...cfz 9157    seqcseq 9573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-fz 9158  df-iseq 9574
This theorem is referenced by:  fisumcvg  10401
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