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Theorem iseraltlem1 13470
Description: Lemma for iseralt 13473. 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 2467 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 eluzelz 11092 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 iseralt.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2575 . . 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 6022 . . . 4  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  RR )
109recnd 9623 . . 3  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  CC )
11 1z 10895 . . 3  |-  1  e.  ZZ
12 uzssz 11102 . . . 4  |-  ( ZZ>= ` 
1 )  C_  ZZ
13 zex 10874 . . . 4  |-  ZZ  e.  _V
1412, 13climconst2 13337 . . 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 11100 . . . 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 6023 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  e.  RR )
20 eluzelz 11092 . . . . 5  |-  ( n  e.  ( ZZ>= `  N
)  ->  n  e.  ZZ )
2120adantl 466 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ZZ )
22 fvex 5876 . . . . 5  |-  ( G `
 N )  e. 
_V
2322fvconst2 6117 . . . 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 2555 . 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 11685 . . . . . 6  |-  ( k  e.  ( N ... n )  ->  k  e.  ( ZZ>= `  N )
)
313uztrn2 11100 . . . . . 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 6023 . . . 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 11685 . . . . 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 12107 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  ( G `  N )
)
4241, 24breqtrrd 4473 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  (
( ZZ  X.  {
( G `  N
) } ) `  n ) )
431, 5, 7, 15, 19, 26, 42climle 13428 1  |-  ( (
ph  /\  N  e.  Z )  ->  0  <_  ( G `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4027   class class class wbr 4447    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    <_ cle 9630    - cmin 9806   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673    ~~> cli 13273
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-rep 4558  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  ax-pre-sup 9571
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-rmo 2822  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-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fl 11898  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-rlim 13278
This theorem is referenced by:  iseraltlem3  13472  iseralt  13473
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