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

Theorem seq3id2 9940
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 9446 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2150 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5305 . . . . . . 7  |-  ( x  =  K  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  K
) )
65eqeq2d 2099 . . . . . 6  |-  ( x  =  K  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 K ) ) )
74, 6imbi12d 232 . . . . 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 228 . . . 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 2150 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5305 . . . . . . 7  |-  ( x  =  n  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  n
) )
1110eqeq2d 2099 . . . . . 6  |-  ( x  =  n  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 n ) ) )
129, 11imbi12d 232 . . . . 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 228 . . . 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 2150 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5305 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2099 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
1714, 16imbi12d 232 . . . . 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 228 . . . 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 2150 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5305 . . . . . . 7  |-  ( x  =  N  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  N
) )
2120eqeq2d 2099 . . . . . 6  |-  ( x  =  N  ->  (
(  seq M (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )
2219, 21imbi12d 232 . . . . 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 228 . . . 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 2089 . . . . 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 9451 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 271 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 367 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 74 . . . . 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 5659 . . . . . 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 5314 . . . . . . . . . 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 2446 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3433adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
35 eluzp1p1 9044 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3635ad2antrl 474 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
37 elfzuz3 9437 . . . . . . . . . . . 12  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3837ad2antll 475 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
39 elfzuzb 9434 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
4036, 38, 39sylanbrc 408 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
4131, 34, 40rspcdva 2727 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4241oveq2d 5668 . . . . . . . 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 5659 . . . . . . . . . . 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 2102 . . . . . . . . . 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 2446 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 seqid2.4 . . . . . . . . . 10  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  e.  S )
4945, 47, 48rspcdva 2727 . . . . . . . . 9  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq M (  .+  ,  F ) `  K
) )
5049adantr 270 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq M ( 
.+  ,  F ) `
 K ) )
5142, 50eqtr2d 2121 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
52 simprl 498 . . . . . . . . 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 270 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
55 uztrn 9035 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5652, 54, 55syl2anc 403 . . . . . . . 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 461 . . . . . . . 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 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6156, 58, 60seq3p1 9884 . . . . . . 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 2102 . . . . . 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 154 . . . . 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 526 . . . 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 9076 . . 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 102    = wceq 1289    e. wcel 1438   A.wral 2359   ` cfv 5015  (class class class)co 5652   1c1 7351    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq 9852
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-fz 9425  df-iseq 9853  df-seq3 9854
This theorem is referenced by:  fsum3cvg  10767
  Copyright terms: Public domain W3C validator