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Theorem iseraltlem1 13586
Description: Lemma for iseralt 13589. 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 2454 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 eluzelz 11091 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 iseralt.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2562 . . 3  |-  ( N  e.  Z  ->  N  e.  ZZ )
54adantl 464 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  N  e.  ZZ )
6 iseralt.5 . . 3  |-  ( ph  ->  G  ~~>  0 )
76adantr 463 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  G  ~~>  0 )
8 iseralt.3 . . . . 5  |-  ( ph  ->  G : Z --> RR )
98ffvelrnda 6007 . . . 4  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  RR )
109recnd 9611 . . 3  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  CC )
11 1z 10890 . . 3  |-  1  e.  ZZ
12 uzssz 11101 . . . 4  |-  ( ZZ>= ` 
1 )  C_  ZZ
13 zex 10869 . . . 4  |-  ZZ  e.  _V
1412, 13climconst2 13453 . . 3  |-  ( ( ( G `  N
)  e.  CC  /\  1  e.  ZZ )  ->  ( ZZ  X.  {
( G `  N
) } )  ~~>  ( G `
 N ) )
1510, 11, 14sylancl 660 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  ( ZZ  X.  { ( G `
 N ) } )  ~~>  ( G `  N ) )
168ad2antrr 723 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  G : Z
--> RR )
173uztrn2 11099 . . . 4  |-  ( ( N  e.  Z  /\  n  e.  ( ZZ>= `  N ) )  ->  n  e.  Z )
1817adantll 711 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  Z )
1916, 18ffvelrnd 6008 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  e.  RR )
20 eluzelz 11091 . . . . 5  |-  ( n  e.  ( ZZ>= `  N
)  ->  n  e.  ZZ )
2120adantl 464 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ZZ )
22 fvex 5858 . . . . 5  |-  ( G `
 N )  e. 
_V
2322fvconst2 6103 . . . 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 463 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  N )  e.  RR )
2624, 25eqeltrd 2542 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ( ZZ  X.  { ( G `
 N ) } ) `  n )  e.  RR )
27 simpr 459 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
2816adantr 463 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  G : Z
--> RR )
29 simplr 753 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  Z )
30 elfzuz 11687 . . . . . 6  |-  ( k  e.  ( N ... n )  ->  k  e.  ( ZZ>= `  N )
)
313uztrn2 11099 . . . . . 6  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
3229, 30, 31syl2an 475 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  k  e.  Z )
3328, 32ffvelrnd 6008 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  ( G `  k )  e.  RR )
34 simpl 455 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ph  /\  N  e.  Z ) )
35 elfzuz 11687 . . . . 5  |-  ( k  e.  ( N ... ( n  -  1
) )  ->  k  e.  ( ZZ>= `  N )
)
3631adantll 711 . . . . . 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 712 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
3936, 38syldan 468 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  ( ZZ>= `  N )
)  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4034, 35, 39syl2an 475 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... ( n  -  1 ) ) )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4127, 33, 40monoord2 12120 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  ( G `  N )
)
4241, 24breqtrrd 4465 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  (
( ZZ  X.  {
( G `  N
) } ) `  n ) )
431, 5, 7, 15, 19, 26, 42climle 13544 1  |-  ( (
ph  /\  N  e.  Z )  ->  0  <_  ( G `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {csn 4016   class class class wbr 4439    X. cxp 4986   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9796   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    ~~> cli 13389
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-rep 4550  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  ax-pre-sup 9559
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-rmo 2812  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-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fl 11910  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394
This theorem is referenced by:  iseraltlem3  13588  iseralt  13589
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