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Theorem climle 12434
Description: Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
Hypotheses
Ref Expression
climadd.1  |-  Z  =  ( ZZ>= `  M )
climadd.2  |-  ( ph  ->  M  e.  ZZ )
climadd.4  |-  ( ph  ->  F  ~~>  A )
climle.5  |-  ( ph  ->  G  ~~>  B )
climle.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
climle.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
climle.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
Assertion
Ref Expression
climle  |-  ( ph  ->  A  <_  B )
Distinct variable groups:    B, k    k, F    ph, k    A, k   
k, G    k, M    k, Z

Proof of Theorem climle
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 climadd.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 climadd.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 climle.5 . . . 4  |-  ( ph  ->  G  ~~>  B )
4 fvex 5743 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2507 . . . . . 6  |-  Z  e. 
_V
65mptex 5967 . . . . 5  |-  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j ) ) )  e.  _V
76a1i 11 . . . 4  |-  ( ph  ->  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) )  e.  _V )
8 climadd.4 . . . 4  |-  ( ph  ->  F  ~~>  A )
9 climle.7 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
109recnd 9115 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
11 climle.6 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
1211recnd 9115 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
13 fveq2 5729 . . . . . . 7  |-  ( j  =  k  ->  ( G `  j )  =  ( G `  k ) )
14 fveq2 5729 . . . . . . 7  |-  ( j  =  k  ->  ( F `  j )  =  ( F `  k ) )
1513, 14oveq12d 6100 . . . . . 6  |-  ( j  =  k  ->  (
( G `  j
)  -  ( F `
 j ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
16 eqid 2437 . . . . . 6  |-  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j ) ) )  =  ( j  e.  Z  |->  ( ( G `
 j )  -  ( F `  j ) ) )
17 ovex 6107 . . . . . 6  |-  ( ( G `  k )  -  ( F `  k ) )  e. 
_V
1815, 16, 17fvmpt 5807 . . . . 5  |-  ( k  e.  Z  ->  (
( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) ) `  k
)  =  ( ( G `  k )  -  ( F `  k ) ) )
1918adantl 454 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) ) `  k
)  =  ( ( G `  k )  -  ( F `  k ) ) )
201, 2, 3, 7, 8, 10, 12, 19climsub 12428 . . 3  |-  ( ph  ->  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) )  ~~>  ( B  -  A ) )
219, 11resubcld 9466 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G `  k
)  -  ( F `
 k ) )  e.  RR )
2219, 21eqeltrd 2511 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) ) `  k
)  e.  RR )
23 climle.8 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
249, 11subge0d 9617 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
0  <_  ( ( G `  k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
2523, 24mpbird 225 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( ( G `  k )  -  ( F `  k )
) )
2625, 19breqtrrd 4239 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( ( j  e.  Z  |->  ( ( G `
 j )  -  ( F `  j ) ) ) `  k
) )
271, 2, 20, 22, 26climge0 12379 . 2  |-  ( ph  ->  0  <_  ( B  -  A ) )
281, 2, 3, 9climrecl 12378 . . 3  |-  ( ph  ->  B  e.  RR )
291, 2, 8, 11climrecl 12378 . . 3  |-  ( ph  ->  A  e.  RR )
3028, 29subge0d 9617 . 2  |-  ( ph  ->  ( 0  <_  ( B  -  A )  <->  A  <_  B ) )
3127, 30mpbid 203 1  |-  ( ph  ->  A  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957   class class class wbr 4213    e. cmpt 4267   ` cfv 5455  (class class class)co 6082   RRcr 8990   0cc0 8991    <_ cle 9122    - cmin 9292   ZZcz 10283   ZZ>=cuz 10489    ~~> cli 12279
This theorem is referenced by:  climlec2  12453  iserle  12454  iseraltlem1  12476  iserabs  12595  cvgcmpub  12597  itg2monolem1  19643  ulmdvlem1  20317  dchrisumlema  21183  dchrisumlem3  21186  stirlinglem10  27809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fl 11203  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284
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