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Theorem climsuselem1 31376
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 2545 . . . 4  |-  ( K  e.  Z  <->  K  e.  ( ZZ>= `  M )
)
32biimpi 194 . . 3  |-  ( K  e.  Z  ->  K  e.  ( ZZ>= `  M )
)
43adantl 466 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
5 simpl 457 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  ph )
6 fveq2 5866 . . . . 5  |-  ( j  =  M  ->  (
I `  j )  =  ( I `  M ) )
7 fveq2 5866 . . . . 5  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
86, 7eleq12d 2549 . . . 4  |-  ( j  =  M  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
98imbi2d 316 . . 3  |-  ( j  =  M  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  M )  e.  ( ZZ>= `  M
) ) ) )
10 fveq2 5866 . . . . 5  |-  ( j  =  k  ->  (
I `  j )  =  ( I `  k ) )
11 fveq2 5866 . . . . 5  |-  ( j  =  k  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  k )
)
1210, 11eleq12d 2549 . . . 4  |-  ( j  =  k  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  k )  e.  (
ZZ>= `  k ) ) )
1312imbi2d 316 . . 3  |-  ( j  =  k  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  k )  e.  ( ZZ>= `  k
) ) ) )
14 fveq2 5866 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
I `  j )  =  ( I `  ( k  +  1 ) ) )
15 fveq2 5866 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( k  +  1 ) ) )
1614, 15eleq12d 2549 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( k  +  1 ) ) ) )
1716imbi2d 316 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
k  +  1 ) ) ) ) )
18 fveq2 5866 . . . . 5  |-  ( j  =  K  ->  (
I `  j )  =  ( I `  K ) )
19 fveq2 5866 . . . . 5  |-  ( j  =  K  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  K )
)
2018, 19eleq12d 2549 . . . 4  |-  ( j  =  K  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  K )  e.  (
ZZ>= `  K ) ) )
2120imbi2d 316 . . 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 2565 . . . 4  |-  ( ph  ->  ( I `  M
)  e.  ( ZZ>= `  M ) )
2423a1i 11 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
25 simpr 461 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ph )
26 simpll 753 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  k  e.  ( ZZ>= `  M ) )
27 simplr 754 . . . . . 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 11092 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
30293ad2ant2 1018 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  ZZ )
3130peano2zd 10970 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  ZZ )
3231zred 10967 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  RR )
33 eluzelre 11093 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  ( I `  k )  e.  RR )
34333ad2ant3 1019 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  k
)  e.  RR )
35 1re 9596 . . . . . . . . 9  |-  1  e.  RR
3635a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
1  e.  RR )
3734, 36readdcld 9624 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  k )  +  1 )  e.  RR )
381eqimss2i 3559 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  Z
3938a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ZZ>= `  M )  C_  Z )
4039sseld 3503 . . . . . . . . . . . 12  |-  ( ph  ->  ( k  e.  (
ZZ>= `  M )  -> 
k  e.  Z ) )
4140imdistani 690 . . . . . . . . . . 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 1016 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( ( I `  k )  +  1 ) ) )
45 eluzelz 11092 . . . . . . . . 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 10967 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  RR )
4830zred 10967 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  RR )
49 eluzle 11095 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  k  <_  ( I `  k ) )
50493ad2ant3 1019 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  <_  ( I `  k ) )
5148, 34, 36, 50leadd1dd 10167 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( (
I `  k )  +  1 ) )
52 eluzle 11095 . . . . . . . 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 9739 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( I `  ( k  +  1 ) ) )
55 eluz 11096 . . . . . . 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 661 . . . . . 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 232 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5825, 26, 28, 57syl3anc 1228 . . . 4  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5958exp31 604 . . 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 11140 . 2  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( I `  K
)  e.  ( ZZ>= `  K ) ) )
614, 5, 60sylc 60 1  |-  ( (
ph  /\  K  e.  Z )  ->  (
I `  K )  e.  ( ZZ>= `  K )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   RRcr 9492   1c1 9494    + caddc 9496    <_ cle 9630   ZZcz 10865   ZZ>=cuz 11083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084
This theorem is referenced by:  climsuse  31377
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