MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iseraltlem1 Structured version   Unicode version

Theorem iseraltlem1 13171
Description: Lemma for iseralt 13174. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
iseralt.1  |-  Z  =  ( ZZ>= `  M )
iseralt.2  |-  ( ph  ->  M  e.  ZZ )
iseralt.3  |-  ( ph  ->  G : Z --> RR )
iseralt.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
iseralt.5  |-  ( ph  ->  G  ~~>  0 )
Assertion
Ref Expression
iseraltlem1  |-  ( (
ph  /\  N  e.  Z )  ->  0  <_  ( G `  N
) )
Distinct variable groups:    k, G    k, M    ph, k    k, N   
k, Z

Proof of Theorem iseraltlem1
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 eluzelz 10882 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 iseralt.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2535 . . 3  |-  ( N  e.  Z  ->  N  e.  ZZ )
54adantl 466 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  N  e.  ZZ )
6 iseralt.5 . . 3  |-  ( ph  ->  G  ~~>  0 )
76adantr 465 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  G  ~~>  0 )
8 iseralt.3 . . . . 5  |-  ( ph  ->  G : Z --> RR )
98ffvelrnda 5855 . . . 4  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  RR )
109recnd 9424 . . 3  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  CC )
11 1z 10688 . . 3  |-  1  e.  ZZ
12 uzssz 10892 . . . 4  |-  ( ZZ>= ` 
1 )  C_  ZZ
13 zex 10667 . . . 4  |-  ZZ  e.  _V
1412, 13climconst2 13038 . . 3  |-  ( ( ( G `  N
)  e.  CC  /\  1  e.  ZZ )  ->  ( ZZ  X.  {
( G `  N
) } )  ~~>  ( G `
 N ) )
1510, 11, 14sylancl 662 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  ( ZZ  X.  { ( G `
 N ) } )  ~~>  ( G `  N ) )
168ad2antrr 725 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  G : Z
--> RR )
173uztrn2 10890 . . . 4  |-  ( ( N  e.  Z  /\  n  e.  ( ZZ>= `  N ) )  ->  n  e.  Z )
1817adantll 713 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  Z )
1916, 18ffvelrnd 5856 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  e.  RR )
20 eluzelz 10882 . . . . 5  |-  ( n  e.  ( ZZ>= `  N
)  ->  n  e.  ZZ )
2120adantl 466 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ZZ )
22 fvex 5713 . . . . 5  |-  ( G `
 N )  e. 
_V
2322fvconst2 5945 . . . 4  |-  ( n  e.  ZZ  ->  (
( ZZ  X.  {
( G `  N
) } ) `  n )  =  ( G `  N ) )
2421, 23syl 16 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ( ZZ  X.  { ( G `
 N ) } ) `  n )  =  ( G `  N ) )
259adantr 465 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  N )  e.  RR )
2624, 25eqeltrd 2517 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ( ZZ  X.  { ( G `
 N ) } ) `  n )  e.  RR )
27 simpr 461 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
2816adantr 465 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  G : Z
--> RR )
29 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  Z )
30 elfzuz 11461 . . . . . 6  |-  ( k  e.  ( N ... n )  ->  k  e.  ( ZZ>= `  N )
)
313uztrn2 10890 . . . . . 6  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
3229, 30, 31syl2an 477 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  k  e.  Z )
3328, 32ffvelrnd 5856 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  ( G `  k )  e.  RR )
34 simpl 457 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ph  /\  N  e.  Z ) )
35 elfzuz 11461 . . . . 5  |-  ( k  e.  ( N ... ( n  -  1
) )  ->  k  e.  ( ZZ>= `  N )
)
3631adantll 713 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
37 iseralt.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
3837adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
3936, 38syldan 470 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  ( ZZ>= `  N )
)  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4034, 35, 39syl2an 477 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... ( n  -  1 ) ) )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4127, 33, 40monoord2 11849 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  ( G `  N )
)
4241, 24breqtrrd 4330 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  (
( ZZ  X.  {
( G `  N
) } ) `  n ) )
431, 5, 7, 15, 19, 26, 42climle 13129 1  |-  ( (
ph  /\  N  e.  Z )  ->  0  <_  ( G `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3889   class class class wbr 4304    X. cxp 4850   -->wf 5426   ` cfv 5430  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    + caddc 9297    <_ cle 9431    - cmin 9607   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449    ~~> cli 12974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fl 11654  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979
This theorem is referenced by:  iseraltlem3  13173  iseralt  13174
  Copyright terms: Public domain W3C validator