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Theorem climsuselem1 27600
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 2468 . . . 4  |-  ( K  e.  Z  <->  K  e.  ( ZZ>= `  M )
)
32biimpi 187 . . 3  |-  ( K  e.  Z  ->  K  e.  ( ZZ>= `  M )
)
43adantl 453 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
5 simpl 444 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  ph )
6 fveq2 5687 . . . . 5  |-  ( j  =  M  ->  (
I `  j )  =  ( I `  M ) )
7 fveq2 5687 . . . . 5  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
86, 7eleq12d 2472 . . . 4  |-  ( j  =  M  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
98imbi2d 308 . . 3  |-  ( j  =  M  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  M )  e.  ( ZZ>= `  M
) ) ) )
10 fveq2 5687 . . . . 5  |-  ( j  =  k  ->  (
I `  j )  =  ( I `  k ) )
11 fveq2 5687 . . . . 5  |-  ( j  =  k  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  k )
)
1210, 11eleq12d 2472 . . . 4  |-  ( j  =  k  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  k )  e.  (
ZZ>= `  k ) ) )
1312imbi2d 308 . . 3  |-  ( j  =  k  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  k )  e.  ( ZZ>= `  k
) ) ) )
14 fveq2 5687 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
I `  j )  =  ( I `  ( k  +  1 ) ) )
15 fveq2 5687 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( k  +  1 ) ) )
1614, 15eleq12d 2472 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( k  +  1 ) ) ) )
1716imbi2d 308 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
k  +  1 ) ) ) ) )
18 fveq2 5687 . . . . 5  |-  ( j  =  K  ->  (
I `  j )  =  ( I `  K ) )
19 fveq2 5687 . . . . 5  |-  ( j  =  K  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  K )
)
2018, 19eleq12d 2472 . . . 4  |-  ( j  =  K  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  K )  e.  (
ZZ>= `  K ) ) )
2120imbi2d 308 . . 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 2494 . . . 4  |-  ( ph  ->  ( I `  M
)  e.  ( ZZ>= `  M ) )
2423a1i 11 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
25 simpr 448 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ph )
26 simpll 731 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  k  e.  ( ZZ>= `  M ) )
27 simplr 732 . . . . . 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 10452 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
30293ad2ant2 979 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  ZZ )
3130peano2zd 10334 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  ZZ )
3231zred 10331 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  RR )
33 eluzelre 10453 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  ( I `  k )  e.  RR )
34333ad2ant3 980 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  k
)  e.  RR )
35 1re 9046 . . . . . . . . 9  |-  1  e.  RR
3635a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
1  e.  RR )
3734, 36readdcld 9071 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  k )  +  1 )  e.  RR )
381eqimss2i 3363 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  Z
3938a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ZZ>= `  M )  C_  Z )
4039sseld 3307 . . . . . . . . . . . 12  |-  ( ph  ->  ( k  e.  (
ZZ>= `  M )  -> 
k  e.  Z ) )
4140imdistani 672 . . . . . . . . . . 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 977 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( ( I `  k )  +  1 ) ) )
45 eluzelz 10452 . . . . . . . . 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 10331 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  RR )
4830zred 10331 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  RR )
49 eluzle 10454 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  k  <_  ( I `  k ) )
50493ad2ant3 980 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  <_  ( I `  k ) )
5148, 34, 36, 50leadd1dd 9596 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( (
I `  k )  +  1 ) )
52 eluzle 10454 . . . . . . . 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 9183 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( I `  ( k  +  1 ) ) )
55 eluz 10455 . . . . . . 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 643 . . . . . 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 224 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5825, 26, 28, 57syl3anc 1184 . . . 4  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5958exp31 588 . . 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 10490 . 2  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( I `  K
)  e.  ( ZZ>= `  K ) ) )
614, 5, 60sylc 58 1  |-  ( (
ph  /\  K  e.  Z )  ->  (
I `  K )  e.  ( ZZ>= `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   RRcr 8945   1c1 8947    + caddc 8949    <_ cle 9077   ZZcz 10238   ZZ>=cuz 10444
This theorem is referenced by:  climsuse  27601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445
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