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Theorem climle 12129
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 5555 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2366 . . . . . 6  |-  Z  e. 
_V
65mptex 5762 . . . . 5  |-  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j ) ) )  e.  _V
76a1i 10 . . . 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 8877 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
11 climle.6 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
1211recnd 8877 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
13 fveq2 5541 . . . . . . 7  |-  ( j  =  k  ->  ( G `  j )  =  ( G `  k ) )
14 fveq2 5541 . . . . . . 7  |-  ( j  =  k  ->  ( F `  j )  =  ( F `  k ) )
1513, 14oveq12d 5892 . . . . . 6  |-  ( j  =  k  ->  (
( G `  j
)  -  ( F `
 j ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
16 eqid 2296 . . . . . 6  |-  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j ) ) )  =  ( j  e.  Z  |->  ( ( G `
 j )  -  ( F `  j ) ) )
17 ovex 5899 . . . . . 6  |-  ( ( G `  k )  -  ( F `  k ) )  e. 
_V
1815, 16, 17fvmpt 5618 . . . . 5  |-  ( k  e.  Z  ->  (
( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) ) `  k
)  =  ( ( G `  k )  -  ( F `  k ) ) )
1918adantl 452 . . . 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 12123 . . 3  |-  ( ph  ->  ( j  e.  Z  |->  ( ( G `  j )  -  ( F `  j )
) )  ~~>  ( B  -  A ) )
219, 11resubcld 9227 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G `  k
)  -  ( F `
 k ) )  e.  RR )
2219, 21eqeltrd 2370 . . 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 9378 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
0  <_  ( ( G `  k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
2523, 24mpbird 223 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( ( G `  k )  -  ( F `  k )
) )
2625, 19breqtrrd 4065 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( ( j  e.  Z  |->  ( ( G `
 j )  -  ( F `  j ) ) ) `  k
) )
271, 2, 20, 22, 26climge0 12074 . 2  |-  ( ph  ->  0  <_  ( B  -  A ) )
281, 2, 3, 9climrecl 12073 . . 3  |-  ( ph  ->  B  e.  RR )
291, 2, 8, 11climrecl 12073 . . 3  |-  ( ph  ->  A  e.  RR )
3028, 29subge0d 9378 . 2  |-  ( ph  ->  ( 0  <_  ( B  -  A )  <->  A  <_  B ) )
3127, 30mpbid 201 1  |-  ( ph  ->  A  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    <_ cle 8884    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246    ~~> cli 11974
This theorem is referenced by:  climlec2  12148  iserle  12149  iseraltlem1  12170  iserabs  12289  cvgcmpub  12291  itg2monolem1  19121  ulmdvlem1  19793  dchrisumlema  20653  dchrisumlem3  20656  stirlinglem10  27935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979
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