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Theorem iseraltlem1 12477
Description: Lemma for iseralt 12480. 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 2438 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 eluzelz 10498 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 iseralt.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2530 . . 3  |-  ( N  e.  Z  ->  N  e.  ZZ )
54adantl 454 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  N  e.  ZZ )
6 iseralt.5 . . 3  |-  ( ph  ->  G  ~~>  0 )
76adantr 453 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  G  ~~>  0 )
8 iseralt.3 . . . . 5  |-  ( ph  ->  G : Z --> RR )
98ffvelrnda 5872 . . . 4  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  RR )
109recnd 9116 . . 3  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  CC )
11 1z 10313 . . 3  |-  1  e.  ZZ
12 uzssz 10507 . . . 4  |-  ( ZZ>= ` 
1 )  C_  ZZ
13 zex 10293 . . . 4  |-  ZZ  e.  _V
1412, 13climconst2 12344 . . 3  |-  ( ( ( G `  N
)  e.  CC  /\  1  e.  ZZ )  ->  ( ZZ  X.  {
( G `  N
) } )  ~~>  ( G `
 N ) )
1510, 11, 14sylancl 645 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  ( ZZ  X.  { ( G `
 N ) } )  ~~>  ( G `  N ) )
168ad2antrr 708 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  G : Z
--> RR )
173uztrn2 10505 . . . 4  |-  ( ( N  e.  Z  /\  n  e.  ( ZZ>= `  N ) )  ->  n  e.  Z )
1817adantll 696 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  Z )
1916, 18ffvelrnd 5873 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  e.  RR )
20 eluzelz 10498 . . . . 5  |-  ( n  e.  ( ZZ>= `  N
)  ->  n  e.  ZZ )
2120adantl 454 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ZZ )
22 fvex 5744 . . . . 5  |-  ( G `
 N )  e. 
_V
2322fvconst2 5949 . . . 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 453 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  N )  e.  RR )
2624, 25eqeltrd 2512 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ( ZZ  X.  { ( G `
 N ) } ) `  n )  e.  RR )
27 simpr 449 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
2816adantr 453 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  G : Z
--> RR )
29 simplr 733 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  Z )
30 elfzuz 11057 . . . . . 6  |-  ( k  e.  ( N ... n )  ->  k  e.  ( ZZ>= `  N )
)
313uztrn2 10505 . . . . . 6  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
3229, 30, 31syl2an 465 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  k  e.  Z )
3328, 32ffvelrnd 5873 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  ( G `  k )  e.  RR )
34 simpl 445 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ph  /\  N  e.  Z ) )
35 elfzuz 11057 . . . . 5  |-  ( k  e.  ( N ... ( n  -  1
) )  ->  k  e.  ( ZZ>= `  N )
)
3631adantll 696 . . . . . 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 697 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
3936, 38syldan 458 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  ( ZZ>= `  N )
)  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4034, 35, 39syl2an 465 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... ( n  -  1 ) ) )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4127, 33, 40monoord2 11356 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  ( G `  N )
)
4241, 24breqtrrd 4240 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  (
( ZZ  X.  {
( G `  N
) } ) `  n ) )
431, 5, 7, 15, 19, 26, 42climle 12435 1  |-  ( (
ph  /\  N  e.  Z )  ->  0  <_  ( G `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {csn 3816   class class class wbr 4214    X. cxp 4878   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    <_ cle 9123    - cmin 9293   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045    ~~> cli 12280
This theorem is referenced by:  iseraltlem3  12479  iseralt  12480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fl 11204  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285
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