Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  climsuselem1 Structured version   Unicode version

Theorem climsuselem1 27747
Description: The subsequence index  I has the expected properties: it belongs to the same upper integers as the original index, and it is always larger or equal than the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
climsuselem1.1  |-  Z  =  ( ZZ>= `  M )
climsuselem1.2  |-  ( ph  ->  M  e.  ZZ )
climsuselem1.3  |-  ( ph  ->  ( I `  M
)  e.  Z )
climsuselem1.4  |-  ( (
ph  /\  k  e.  Z )  ->  (
I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) ) )
Assertion
Ref Expression
climsuselem1  |-  ( (
ph  /\  K  e.  Z )  ->  (
I `  K )  e.  ( ZZ>= `  K )
)
Distinct variable groups:    ph, k    k, I    k, M    k, Z
Allowed substitution hint:    K( k)

Proof of Theorem climsuselem1
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 climsuselem1.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
21eleq2i 2506 . . . 4  |-  ( K  e.  Z  <->  K  e.  ( ZZ>= `  M )
)
32biimpi 188 . . 3  |-  ( K  e.  Z  ->  K  e.  ( ZZ>= `  M )
)
43adantl 454 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
5 simpl 445 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  ph )
6 fveq2 5757 . . . . 5  |-  ( j  =  M  ->  (
I `  j )  =  ( I `  M ) )
7 fveq2 5757 . . . . 5  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
86, 7eleq12d 2510 . . . 4  |-  ( j  =  M  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
98imbi2d 309 . . 3  |-  ( j  =  M  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  M )  e.  ( ZZ>= `  M
) ) ) )
10 fveq2 5757 . . . . 5  |-  ( j  =  k  ->  (
I `  j )  =  ( I `  k ) )
11 fveq2 5757 . . . . 5  |-  ( j  =  k  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  k )
)
1210, 11eleq12d 2510 . . . 4  |-  ( j  =  k  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  k )  e.  (
ZZ>= `  k ) ) )
1312imbi2d 309 . . 3  |-  ( j  =  k  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  k )  e.  ( ZZ>= `  k
) ) ) )
14 fveq2 5757 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
I `  j )  =  ( I `  ( k  +  1 ) ) )
15 fveq2 5757 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( k  +  1 ) ) )
1614, 15eleq12d 2510 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( k  +  1 ) ) ) )
1716imbi2d 309 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
k  +  1 ) ) ) ) )
18 fveq2 5757 . . . . 5  |-  ( j  =  K  ->  (
I `  j )  =  ( I `  K ) )
19 fveq2 5757 . . . . 5  |-  ( j  =  K  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  K )
)
2018, 19eleq12d 2510 . . . 4  |-  ( j  =  K  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  K )  e.  (
ZZ>= `  K ) ) )
2120imbi2d 309 . . 3  |-  ( j  =  K  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  K )  e.  ( ZZ>= `  K
) ) ) )
22 climsuselem1.3 . . . . 5  |-  ( ph  ->  ( I `  M
)  e.  Z )
2322, 1syl6eleq 2532 . . . 4  |-  ( ph  ->  ( I `  M
)  e.  ( ZZ>= `  M ) )
2423a1i 11 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
25 simpr 449 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ph )
26 simpll 732 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  k  e.  ( ZZ>= `  M ) )
27 simplr 733 . . . . . 6  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ( ph  ->  (
I `  k )  e.  ( ZZ>= `  k )
) )
2825, 27mpd 15 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ( I `  k
)  e.  ( ZZ>= `  k ) )
29 eluzelz 10527 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
30293ad2ant2 980 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  ZZ )
3130peano2zd 10409 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  ZZ )
3231zred 10406 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  RR )
33 eluzelre 10528 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  ( I `  k )  e.  RR )
34333ad2ant3 981 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  k
)  e.  RR )
35 1re 9121 . . . . . . . . 9  |-  1  e.  RR
3635a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
1  e.  RR )
3734, 36readdcld 9146 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  k )  +  1 )  e.  RR )
381eqimss2i 3389 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  Z
3938a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ZZ>= `  M )  C_  Z )
4039sseld 3333 . . . . . . . . . . . 12  |-  ( ph  ->  ( k  e.  (
ZZ>= `  M )  -> 
k  e.  Z ) )
4140imdistani 673 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ph  /\  k  e.  Z ) )
42 climsuselem1.4 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  (
I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) ) )
4341, 42syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( ( I `
 k )  +  1 ) ) )
44433adant3 978 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( ( I `  k )  +  1 ) ) )
45 eluzelz 10527 . . . . . . . . 9  |-  ( ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) )  ->  ( I `  ( k  +  1 ) )  e.  ZZ )
4644, 45syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ZZ )
4746zred 10406 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  RR )
4830zred 10406 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  RR )
49 eluzle 10529 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  k  <_  ( I `  k ) )
50493ad2ant3 981 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  <_  ( I `  k ) )
5148, 34, 36, 50leadd1dd 9671 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( (
I `  k )  +  1 ) )
52 eluzle 10529 . . . . . . . 8  |-  ( ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) )  ->  ( (
I `  k )  +  1 )  <_ 
( I `  (
k  +  1 ) ) )
5344, 52syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  k )  +  1 )  <_  ( I `  ( k  +  1 ) ) )
5432, 37, 47, 51, 53letrd 9258 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( I `  ( k  +  1 ) ) )
55 eluz 10530 . . . . . . 7  |-  ( ( ( k  +  1 )  e.  ZZ  /\  ( I `  (
k  +  1 ) )  e.  ZZ )  ->  ( ( I `
 ( k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) )  <-> 
( k  +  1 )  <_  ( I `  ( k  +  1 ) ) ) )
5631, 46, 55syl2anc 644 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( k  +  1 ) )  <->  ( k  +  1 )  <_ 
( I `  (
k  +  1 ) ) ) )
5754, 56mpbird 225 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5825, 26, 28, 57syl3anc 1185 . . . 4  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5958exp31 589 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) )  ->  ( ph  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
k  +  1 ) ) ) ) )
609, 13, 17, 21, 24, 59uzind4 10565 . 2  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( I `  K
)  e.  ( ZZ>= `  K ) ) )
614, 5, 60sylc 59 1  |-  ( (
ph  /\  K  e.  Z )  ->  (
I `  K )  e.  ( ZZ>= `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    C_ wss 3306   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   RRcr 9020   1c1 9022    + caddc 9024    <_ cle 9152   ZZcz 10313   ZZ>=cuz 10519
This theorem is referenced by:  climsuse  27748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520
  Copyright terms: Public domain W3C validator