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Theorem iseraltlem1 12170
Description: Lemma for iseralt 12173. 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 2296 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 eluzelz 10254 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 iseralt.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3eleq2s 2388 . . 3  |-  ( N  e.  Z  ->  N  e.  ZZ )
54adantl 452 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  N  e.  ZZ )
6 iseralt.5 . . 3  |-  ( ph  ->  G  ~~>  0 )
76adantr 451 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  G  ~~>  0 )
8 iseralt.3 . . . . 5  |-  ( ph  ->  G : Z --> RR )
9 ffvelrn 5679 . . . . 5  |-  ( ( G : Z --> RR  /\  N  e.  Z )  ->  ( G `  N
)  e.  RR )
108, 9sylan 457 . . . 4  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  RR )
1110recnd 8877 . . 3  |-  ( (
ph  /\  N  e.  Z )  ->  ( G `  N )  e.  CC )
12 1z 10069 . . 3  |-  1  e.  ZZ
13 uzssz 10263 . . . 4  |-  ( ZZ>= ` 
1 )  C_  ZZ
14 zex 10049 . . . 4  |-  ZZ  e.  _V
1513, 14climconst2 12038 . . 3  |-  ( ( ( G `  N
)  e.  CC  /\  1  e.  ZZ )  ->  ( ZZ  X.  {
( G `  N
) } )  ~~>  ( G `
 N ) )
1611, 12, 15sylancl 643 . 2  |-  ( (
ph  /\  N  e.  Z )  ->  ( ZZ  X.  { ( G `
 N ) } )  ~~>  ( G `  N ) )
178ad2antrr 706 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  G : Z
--> RR )
183uztrn2 10261 . . . 4  |-  ( ( N  e.  Z  /\  n  e.  ( ZZ>= `  N ) )  ->  n  e.  Z )
1918adantll 694 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  Z )
20 ffvelrn 5679 . . 3  |-  ( ( G : Z --> RR  /\  n  e.  Z )  ->  ( G `  n
)  e.  RR )
2117, 19, 20syl2anc 642 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  e.  RR )
22 eluzelz 10254 . . . . 5  |-  ( n  e.  ( ZZ>= `  N
)  ->  n  e.  ZZ )
2322adantl 452 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ZZ )
24 fvex 5555 . . . . 5  |-  ( G `
 N )  e. 
_V
2524fvconst2 5745 . . . 4  |-  ( n  e.  ZZ  ->  (
( ZZ  X.  {
( G `  N
) } ) `  n )  =  ( G `  N ) )
2623, 25syl 15 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ( ZZ  X.  { ( G `
 N ) } ) `  n )  =  ( G `  N ) )
2710adantr 451 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  N )  e.  RR )
2826, 27eqeltrd 2370 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ( ZZ  X.  { ( G `
 N ) } ) `  n )  e.  RR )
29 simpr 447 . . . 4  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  n  e.  ( ZZ>= `  N )
)
3017adantr 451 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  G : Z
--> RR )
31 simplr 731 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  N  e.  Z )
32 elfzuz 10810 . . . . . 6  |-  ( k  e.  ( N ... n )  ->  k  e.  ( ZZ>= `  N )
)
333uztrn2 10261 . . . . . 6  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
3431, 32, 33syl2an 463 . . . . 5  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  k  e.  Z )
35 ffvelrn 5679 . . . . 5  |-  ( ( G : Z --> RR  /\  k  e.  Z )  ->  ( G `  k
)  e.  RR )
3630, 34, 35syl2anc 642 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... n ) )  ->  ( G `  k )  e.  RR )
37 simpl 443 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( ph  /\  N  e.  Z ) )
38 elfzuz 10810 . . . . 5  |-  ( k  e.  ( N ... ( n  -  1
) )  ->  k  e.  ( ZZ>= `  N )
)
3933adantll 694 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
40 iseralt.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
4140adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  Z )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k ) )
4239, 41syldan 456 . . . . 5  |-  ( ( ( ph  /\  N  e.  Z )  /\  k  e.  ( ZZ>= `  N )
)  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4337, 38, 42syl2an 463 . . . 4  |-  ( ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>=
`  N ) )  /\  k  e.  ( N ... ( n  -  1 ) ) )  ->  ( G `  ( k  +  1 ) )  <_  ( G `  k )
)
4429, 36, 43monoord2 11093 . . 3  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  ( G `  N )
)
4544, 26breqtrrd 4065 . 2  |-  ( ( ( ph  /\  N  e.  Z )  /\  n  e.  ( ZZ>= `  N )
)  ->  ( G `  n )  <_  (
( ZZ  X.  {
( G `  N
) } ) `  n ) )
461, 5, 7, 16, 21, 28, 45climle 12129 1  |-  ( (
ph  /\  N  e.  Z )  ->  0  <_  ( G `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    ~~> cli 11974
This theorem is referenced by:  iseraltlem3  12172  iseralt  12173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979
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