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Theorem climub 11068
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
clim2iser.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 2117 . 2  |-  ( ZZ>= `  N )  =  (
ZZ>= `  N )
2 climub.2 . . . 4  |-  ( ph  ->  N  e.  Z )
3 clim2iser.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
42, 3eleqtrdi 2210 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9291 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . 2  |-  ( ph  ->  N  e.  ZZ )
7 fveq2 5389 . . . . . 6  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
87eleq1d 2186 . . . . 5  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
98imbi2d 229 . . . 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 115 . . . 4  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
129, 11vtoclga 2726 . . 3  |-  ( N  e.  Z  ->  ( ph  ->  ( F `  N )  e.  RR ) )
132, 12mpcom 36 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR )
14 climub.3 . 2  |-  ( ph  ->  F  ~~>  A )
153uztrn2 9299 . . . 4  |-  ( ( N  e.  Z  /\  j  e.  ( ZZ>= `  N ) )  -> 
j  e.  Z )
162, 15sylan 281 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  Z )
17 fveq2 5389 . . . . . . 7  |-  ( k  =  j  ->  ( F `  k )  =  ( F `  j ) )
1817eleq1d 2186 . . . . . 6  |-  ( k  =  j  ->  (
( F `  k
)  e.  RR  <->  ( F `  j )  e.  RR ) )
1918imbi2d 229 . . . . 5  |-  ( k  =  j  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  j
)  e.  RR ) ) )
2019, 11vtoclga 2726 . . . 4  |-  ( j  e.  Z  ->  ( ph  ->  ( F `  j )  e.  RR ) )
2120impcom 124 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  e.  RR )
2216, 21syldan 280 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  j )  e.  RR )
23 simpr 109 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  j  e.  ( ZZ>= `  N )
)
24 elfzuz 9757 . . . . 5  |-  ( k  e.  ( N ... j )  ->  k  e.  ( ZZ>= `  N )
)
253uztrn2 9299 . . . . . . 7  |-  ( ( N  e.  Z  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  Z )
262, 25sylan 281 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
2726, 10syldan 280 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  e.  RR )
2824, 27sylan2 284 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
2928adantlr 468 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... j ) )  ->  ( F `  k )  e.  RR )
30 elfzuz 9757 . . . . 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 280 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3330, 32sylan2 284 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3433adantlr 468 . . 3  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  N )
)  /\  k  e.  ( N ... ( j  -  1 ) ) )  ->  ( F `  k )  <_  ( F `  ( k  +  1 ) ) )
3523, 29, 34monoord 10204 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  N )
)  ->  ( F `  N )  <_  ( F `  j )
)
361, 6, 13, 14, 22, 35climlec2 11065 1  |-  ( ph  ->  ( F `  N
)  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   1c1 7589    + caddc 7591    <_ cle 7769    - cmin 7901   ZZcz 9012   ZZ>=cuz 9282   ...cfz 9745    ~~> cli 11002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-rp 9398  df-fz 9746  df-seqfrec 10174  df-exp 10248  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-clim 11003
This theorem is referenced by:  climserle  11069
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