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Theorem climub 12414
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 2408 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 climub.2 . . . 4  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2498 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 10456 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 16 . 2  |-  ( ph  ->  N  e.  ZZ )
7 fveq2 5691 . . . . . 6  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
87eleq1d 2474 . . . . 5  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
98imbi2d 308 . . . 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 425 . . . 4  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
129, 11vtoclga 2981 . . 3  |-  ( N  e.  Z  ->  ( ph  ->  ( F `  N )  e.  RR ) )
132, 12mpcom 34 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR )
14 climub.3 . 2  |-  ( ph  ->  F  ~~>  A )
153uztrn2 10463 . . . 4  |-  ( ( N  e.  Z  /\  j  e.  ( ZZ>= `  N ) )  -> 
j  e.  Z )
162, 15sylan 458 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  Z )
17 fveq2 5691 . . . . . . 7  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1817eleq1d 2474 . . . . . 6  |-  ( k  =  j  ->  (
( F `  k
)  e.  RR  <->  ( F `  j )  e.  RR ) )
1918imbi2d 308 . . . . 5  |-  ( k  =  j  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  j
)  e.  RR ) ) )
2019, 11vtoclga 2981 . . . 4  |-  ( j  e.  Z  ->  ( ph  ->  ( F `  j )  e.  RR ) )
2120impcom 420 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
2216, 21syldan 457 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  j )  e.  RR )
23 simpr 448 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  ( ZZ>= `  N )
)
24 elfzuz 11015 . . . . 5  |-  ( k  e.  ( N ... j )  ->  k  e.  ( ZZ>= `  N )
)
253uztrn2 10463 . . . . . . 7  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
262, 25sylan 458 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
2726, 10syldan 457 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  e.  RR )
2824, 27sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
2928adantlr 696 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
30 elfzuz 11015 . . . . 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 457 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3330, 32sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3433adantlr 696 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3523, 29, 34monoord 11312 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  N )  <_  ( F `  j )
)
361, 6, 13, 14, 22, 35climlec2 12411 1  |-  ( ph  ->  ( F `  N
)  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   RRcr 8949   1c1 8951    + caddc 8953    <_ cle 9081    - cmin 9251   ZZcz 10242   ZZ>=cuz 10448   ...cfz 11003    ~~> cli 12237
This theorem is referenced by:  climserle  12415  itg2i1fseqle  19603  emcllem7  20797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fl 11161  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242
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