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Theorem climsuselem1 31852
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 2532 . . . 4  |-  ( K  e.  Z  <->  K  e.  ( ZZ>= `  M )
)
32biimpi 194 . . 3  |-  ( K  e.  Z  ->  K  e.  ( ZZ>= `  M )
)
43adantl 464 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  K  e.  ( ZZ>= `  M )
)
5 simpl 455 . 2  |-  ( (
ph  /\  K  e.  Z )  ->  ph )
6 fveq2 5848 . . . . 5  |-  ( j  =  M  ->  (
I `  j )  =  ( I `  M ) )
7 fveq2 5848 . . . . 5  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
86, 7eleq12d 2536 . . . 4  |-  ( j  =  M  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
98imbi2d 314 . . 3  |-  ( j  =  M  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  M )  e.  ( ZZ>= `  M
) ) ) )
10 fveq2 5848 . . . . 5  |-  ( j  =  k  ->  (
I `  j )  =  ( I `  k ) )
11 fveq2 5848 . . . . 5  |-  ( j  =  k  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  k )
)
1210, 11eleq12d 2536 . . . 4  |-  ( j  =  k  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  k )  e.  (
ZZ>= `  k ) ) )
1312imbi2d 314 . . 3  |-  ( j  =  k  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  k )  e.  ( ZZ>= `  k
) ) ) )
14 fveq2 5848 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
I `  j )  =  ( I `  ( k  +  1 ) ) )
15 fveq2 5848 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( k  +  1 ) ) )
1614, 15eleq12d 2536 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( k  +  1 ) ) ) )
1716imbi2d 314 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( ph  ->  ( I `
 j )  e.  ( ZZ>= `  j )
)  <->  ( ph  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
k  +  1 ) ) ) ) )
18 fveq2 5848 . . . . 5  |-  ( j  =  K  ->  (
I `  j )  =  ( I `  K ) )
19 fveq2 5848 . . . . 5  |-  ( j  =  K  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  K )
)
2018, 19eleq12d 2536 . . . 4  |-  ( j  =  K  ->  (
( I `  j
)  e.  ( ZZ>= `  j )  <->  ( I `  K )  e.  (
ZZ>= `  K ) ) )
2120imbi2d 314 . . 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 2552 . . . 4  |-  ( ph  ->  ( I `  M
)  e.  ( ZZ>= `  M ) )
2423a1i 11 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( I `  M )  e.  (
ZZ>= `  M ) ) )
25 simpr 459 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ph )
26 simpll 751 . . . . 5  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  k  e.  ( ZZ>= `  M ) )
27 simplr 753 . . . . . 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 11091 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
30293ad2ant2 1016 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  ZZ )
3130peano2zd 10968 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  ZZ )
3231zred 10965 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  e.  RR )
33 eluzelre 11092 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  ( I `  k )  e.  RR )
34333ad2ant3 1017 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  k
)  e.  RR )
35 1red 9600 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
1  e.  RR )
3634, 35readdcld 9612 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  k )  +  1 )  e.  RR )
371eqimss2i 3544 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  Z
3837a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ZZ>= `  M )  C_  Z )
3938sseld 3488 . . . . . . . . . . . 12  |-  ( ph  ->  ( k  e.  (
ZZ>= `  M )  -> 
k  e.  Z ) )
4039imdistani 688 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ph  /\  k  e.  Z ) )
41 climsuselem1.4 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  (
I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) ) )
4240, 41syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( ( I `
 k )  +  1 ) ) )
43423adant3 1014 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( ( I `  k )  +  1 ) ) )
44 eluzelz 11091 . . . . . . . . 9  |-  ( ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) )  ->  ( I `  ( k  +  1 ) )  e.  ZZ )
4543, 44syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ZZ )
4645zred 10965 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  RR )
4730zred 10965 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  e.  RR )
48 eluzle 11094 . . . . . . . . 9  |-  ( ( I `  k )  e.  ( ZZ>= `  k
)  ->  k  <_  ( I `  k ) )
49483ad2ant3 1017 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
k  <_  ( I `  k ) )
5047, 34, 35, 49leadd1dd 10162 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( (
I `  k )  +  1 ) )
51 eluzle 11094 . . . . . . . 8  |-  ( ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
( I `  k
)  +  1 ) )  ->  ( (
I `  k )  +  1 )  <_ 
( I `  (
k  +  1 ) ) )
5243, 51syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  k )  +  1 )  <_  ( I `  ( k  +  1 ) ) )
5332, 36, 46, 50, 52letrd 9728 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( k  +  1 )  <_  ( I `  ( k  +  1 ) ) )
54 eluz 11095 . . . . . . 7  |-  ( ( ( k  +  1 )  e.  ZZ  /\  ( I `  (
k  +  1 ) )  e.  ZZ )  ->  ( ( I `
 ( k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) )  <-> 
( k  +  1 )  <_  ( I `  ( k  +  1 ) ) ) )
5531, 45, 54syl2anc 659 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( ( I `  ( k  +  1 ) )  e.  (
ZZ>= `  ( k  +  1 ) )  <->  ( k  +  1 )  <_ 
( I `  (
k  +  1 ) ) ) )
5653, 55mpbird 232 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )  /\  ( I `  k
)  e.  ( ZZ>= `  k ) )  -> 
( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5725, 26, 28, 56syl3anc 1226 . . . 4  |-  ( ( ( k  e.  (
ZZ>= `  M )  /\  ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) ) )  /\  ph )  ->  ( I `  (
k  +  1 ) )  e.  ( ZZ>= `  ( k  +  1 ) ) )
5857exp31 602 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ( ph  ->  ( I `  k )  e.  (
ZZ>= `  k ) )  ->  ( ph  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  (
k  +  1 ) ) ) ) )
599, 13, 17, 21, 24, 58uzind4 11140 . 2  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( I `  K
)  e.  ( ZZ>= `  K ) ) )
604, 5, 59sylc 60 1  |-  ( (
ph  /\  K  e.  Z )  ->  (
I `  K )  e.  ( ZZ>= `  K )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482    + caddc 9484    <_ cle 9618   ZZcz 10860   ZZ>=cuz 11082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083
This theorem is referenced by:  climsuse  31853
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