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Theorem climub 12342
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 2366 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 climub.2 . . . 4  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2456 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 10389 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 15 . 2  |-  ( ph  ->  N  e.  ZZ )
7 fveq2 5632 . . . . . 6  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
87eleq1d 2432 . . . . 5  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
98imbi2d 307 . . . 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 424 . . . 4  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
129, 11vtoclga 2934 . . 3  |-  ( N  e.  Z  ->  ( ph  ->  ( F `  N )  e.  RR ) )
132, 12mpcom 32 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR )
14 climub.3 . 2  |-  ( ph  ->  F  ~~>  A )
153uztrn2 10396 . . . 4  |-  ( ( N  e.  Z  /\  j  e.  ( ZZ>= `  N ) )  -> 
j  e.  Z )
162, 15sylan 457 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  Z )
17 fveq2 5632 . . . . . . 7  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1817eleq1d 2432 . . . . . 6  |-  ( k  =  j  ->  (
( F `  k
)  e.  RR  <->  ( F `  j )  e.  RR ) )
1918imbi2d 307 . . . . 5  |-  ( k  =  j  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  j
)  e.  RR ) ) )
2019, 11vtoclga 2934 . . . 4  |-  ( j  e.  Z  ->  ( ph  ->  ( F `  j )  e.  RR ) )
2120impcom 419 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
2216, 21syldan 456 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  j )  e.  RR )
23 simpr 447 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  ( ZZ>= `  N )
)
24 elfzuz 10947 . . . . 5  |-  ( k  e.  ( N ... j )  ->  k  e.  ( ZZ>= `  N )
)
253uztrn2 10396 . . . . . . 7  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
262, 25sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
2726, 10syldan 456 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  e.  RR )
2824, 27sylan2 460 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
2928adantlr 695 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
30 elfzuz 10947 . . . . 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 456 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3330, 32sylan2 460 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3433adantlr 695 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3523, 29, 34monoord 11240 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  N )  <_  ( F `  j )
)
361, 6, 13, 14, 22, 35climlec2 12339 1  |-  ( ph  ->  ( F `  N
)  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   RRcr 8883   1c1 8885    + caddc 8887    <_ cle 9015    - cmin 9184   ZZcz 10175   ZZ>=cuz 10381   ...cfz 10935    ~~> cli 12165
This theorem is referenced by:  climserle  12343  itg2i1fseqle  19324  emcllem7  20518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fz 10936  df-fl 11089  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170
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