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Theorem climub 12460
Description: The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
climub.2  |-  ( ph  ->  N  e.  Z )
climub.3  |-  ( ph  ->  F  ~~>  A )
climub.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
climub.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( F `  (
k  +  1 ) ) )
Assertion
Ref Expression
climub  |-  ( ph  ->  ( F `  N
)  <_  A )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem climub
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 climub.2 . . . 4  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2528 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 10501 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 16 . 2  |-  ( ph  ->  N  e.  ZZ )
7 fveq2 5731 . . . . . 6  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
87eleq1d 2504 . . . . 5  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
98imbi2d 309 . . . 4  |-  ( k  =  N  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  N
)  e.  RR ) ) )
10 climub.4 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
1110expcom 426 . . . 4  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
129, 11vtoclga 3019 . . 3  |-  ( N  e.  Z  ->  ( ph  ->  ( F `  N )  e.  RR ) )
132, 12mpcom 35 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR )
14 climub.3 . 2  |-  ( ph  ->  F  ~~>  A )
153uztrn2 10508 . . . 4  |-  ( ( N  e.  Z  /\  j  e.  ( ZZ>= `  N ) )  -> 
j  e.  Z )
162, 15sylan 459 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  Z )
17 fveq2 5731 . . . . . . 7  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1817eleq1d 2504 . . . . . 6  |-  ( k  =  j  ->  (
( F `  k
)  e.  RR  <->  ( F `  j )  e.  RR ) )
1918imbi2d 309 . . . . 5  |-  ( k  =  j  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  j
)  e.  RR ) ) )
2019, 11vtoclga 3019 . . . 4  |-  ( j  e.  Z  ->  ( ph  ->  ( F `  j )  e.  RR ) )
2120impcom 421 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
2216, 21syldan 458 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  j )  e.  RR )
23 simpr 449 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  ( ZZ>= `  N )
)
24 elfzuz 11060 . . . . 5  |-  ( k  e.  ( N ... j )  ->  k  e.  ( ZZ>= `  N )
)
253uztrn2 10508 . . . . . . 7  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
262, 25sylan 459 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
2726, 10syldan 458 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  e.  RR )
2824, 27sylan2 462 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
2928adantlr 697 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
30 elfzuz 11060 . . . . 5  |-  ( k  e.  ( N ... ( j  -  1 ) )  ->  k  e.  ( ZZ>= `  N )
)
31 climub.5 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( F `  (
k  +  1 ) ) )
3226, 31syldan 458 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3330, 32sylan2 462 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3433adantlr 697 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3523, 29, 34monoord 11358 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  N )  <_  ( F `  j )
)
361, 6, 13, 14, 22, 35climlec2 12457 1  |-  ( ph  ->  ( F `  N
)  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   RRcr 8994   1c1 8996    + caddc 8998    <_ cle 9126    - cmin 9296   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    ~~> cli 12283
This theorem is referenced by:  climserle  12461  itg2i1fseqle  19649  emcllem7  20845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fl 11207  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288
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