ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seq3id2 Unicode version

Theorem seq3id2 10237
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 9767 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2180 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5389 . . . . . . 7  |-  ( x  =  K  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  K
) )
65eqeq2d 2129 . . . . . 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 2180 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5389 . . . . . . 7  |-  ( x  =  n  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  n
) )
1110eqeq2d 2129 . . . . . 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 2180 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5389 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2129 . . . . . 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 2180 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5389 . . . . . . 7  |-  ( x  =  N  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  N
) )
2120eqeq2d 2129 . . . . . 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 2118 . . . . 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 9772 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 275 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 372 . . . . . 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 5749 . . . . . 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 5398 . . . . . . . . . 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 2482 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3433adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
35 eluzp1p1 9307 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3635ad2antrl 481 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
37 elfzuz3 9758 . . . . . . . . . . . 12  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3837ad2antll 482 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
39 elfzuzb 9755 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
4036, 38, 39sylanbrc 413 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
4131, 34, 40rspcdva 2768 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4241oveq2d 5758 . . . . . . . 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 5749 . . . . . . . . . . 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 2132 . . . . . . . . . 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 2482 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 seqid2.4 . . . . . . . . . 10  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  e.  S )
4945, 47, 48rspcdva 2768 . . . . . . . . 9  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq M (  .+  ,  F ) `  K
) )
5049adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq M ( 
.+  ,  F ) `
 K ) )
5142, 50eqtr2d 2151 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
52 simprl 505 . . . . . . . . 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 274 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
55 uztrn 9298 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5652, 54, 55syl2anc 408 . . . . . . . 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 468 . . . . . . . 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 468 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6156, 58, 60seq3p1 10190 . . . . . . 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 2132 . . . . . 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 533 . . . 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 9339 . . 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 1316    e. wcel 1465   A.wral 2393   ` cfv 5093  (class class class)co 5742   1c1 7589    + caddc 7591   ZZcz 9012   ZZ>=cuz 9282   ...cfz 9745    seqcseq 10173
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-fz 9746  df-seqfrec 10174
This theorem is referenced by:  seq3coll  10540  fsum3cvg  11101
  Copyright terms: Public domain W3C validator