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Theorem seq3id2 10233
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 Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
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 )
seq3id2.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seq3id2.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
seq3id2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  (  seq M ( 
.+  ,  F ) `
 N ) )
Distinct variable groups:    x, y, F   
x, K, y    x, M, y    x, N, y    ph, x, y    x, S, y    x,  .+ , y    x, Z
Allowed substitution hint:    Z( y)

Proof of Theorem seq3id2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 seqid2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 9763 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2178 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5387 . . . . . . 7  |-  ( x  =  K  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  K
) )
65eqeq2d 2127 . . . . . 6  |-  ( x  =  K  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 K ) ) )
74, 6imbi12d 233 . . . . 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 229 . . . 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 2178 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5387 . . . . . . 7  |-  ( x  =  n  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  n
) )
1110eqeq2d 2127 . . . . . 6  |-  ( x  =  n  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 n ) ) )
129, 11imbi12d 233 . . . . 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 229 . . . 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 2178 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5387 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2127 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
1714, 16imbi12d 233 . . . . 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 229 . . . 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 2178 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5387 . . . . . . 7  |-  ( x  =  N  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  N
) )
2120eqeq2d 2127 . . . . . 6  |-  ( x  =  N  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )
2219, 21imbi12d 233 . . . . 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 229 . . . 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 2116 . . . . 5  |-  ( K  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
25242a1i 27 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) ) ) )
26 peano2fzr 9768 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 273 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 370 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 75 . . . . 5  |-  ( (
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 5747 . . . . . 6  |-  ( (  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  n
)  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) )
31 fveqeq2 5396 . . . . . . . . . 10  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
32 seqid2.5 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
3332ralrimiva 2480 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3433adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
35 eluzp1p1 9303 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3635ad2antrl 479 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
37 elfzuz3 9754 . . . . . . . . . . . 12  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3837ad2antll 480 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
39 elfzuzb 9751 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
4036, 38, 39sylanbrc 411 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
4131, 34, 40rspcdva 2766 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4241oveq2d 5756 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq M
(  .+  ,  F
) `  K )  .+  Z ) )
43 oveq1 5747 . . . . . . . . . . 11  |-  ( x  =  (  seq M
(  .+  ,  F
) `  K )  ->  ( x  .+  Z
)  =  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )
)
44 id 19 . . . . . . . . . . 11  |-  ( x  =  (  seq M
(  .+  ,  F
) `  K )  ->  x  =  (  seq M (  .+  ,  F ) `  K
) )
4543, 44eqeq12d 2130 . . . . . . . . . 10  |-  ( x  =  (  seq M
(  .+  ,  F
) `  K )  ->  ( ( x  .+  Z )  =  x  <-> 
( (  seq M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq M (  .+  ,  F ) `  K
) ) )
46 seqid2.1 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
4746ralrimiva 2480 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 seqid2.4 . . . . . . . . . 10  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  e.  S )
4945, 47, 48rspcdva 2766 . . . . . . . . 9  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq M (  .+  ,  F ) `  K
) )
5049adantr 272 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq M ( 
.+  ,  F ) `
 K ) )
5142, 50eqtr2d 2149 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
52 simprl 503 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  K )
)
53 seqid2.2 . . . . . . . . . 10  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
5453adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
55 uztrn 9294 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5652, 54, 55syl2anc 406 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
57 seq3id2.f . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
5857adantlr 466 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  M ) )  ->  ( F `  x )  e.  S
)
59 seq3id2.cl . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6059adantlr 466 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6156, 58, 60seq3p1 10186 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6251, 61eqeq12d 2130 . . . . . 6  |-  ( (
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 155 . . . . 5  |-  ( (
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 ) ) ) )
6429, 63animpimp2impd 531 . . . 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 ) ) ) ) ) )
658, 13, 18, 23, 25, 64uzind4 9335 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) ) )
661, 65mpcom 36 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )
673, 66mpd 13 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  (  seq M ( 
.+  ,  F ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   ` cfv 5091  (class class class)co 5740   1c1 7585    + caddc 7587   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741    seqcseq 10169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742  df-seqfrec 10170
This theorem is referenced by:  seq3coll  10536  fsum3cvg  11097
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