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Theorem rlimsqz 12435
Description: Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
Hypotheses
Ref Expression
rlimsqz.d  |-  ( ph  ->  D  e.  RR )
rlimsqz.m  |-  ( ph  ->  M  e.  RR )
rlimsqz.l  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D
)
rlimsqz.b  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
rlimsqz.c  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
rlimsqz.1  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  <_  C )
rlimsqz.2  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  D )
Assertion
Ref Expression
rlimsqz  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Distinct variable groups:    x, A    x, D    x, M    ph, x
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem rlimsqz
StepHypRef Expression
1 rlimsqz.m . 2  |-  ( ph  ->  M  e.  RR )
2 rlimsqz.d . . 3  |-  ( ph  ->  D  e.  RR )
32recnd 9106 . 2  |-  ( ph  ->  D  e.  CC )
4 rlimsqz.l . 2  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D
)
5 rlimsqz.b . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
65recnd 9106 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
7 rlimsqz.c . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
87recnd 9106 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
95adantrr 698 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  e.  RR )
107adantrr 698 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  e.  RR )
112adantr 452 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  e.  RR )
12 rlimsqz.1 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  <_  C )
139, 10, 11, 12lesub2dd 9635 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( D  -  C
)  <_  ( D  -  B ) )
14 rlimsqz.2 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  D )
1510, 11, 14abssuble0d 12227 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  =  ( D  -  C ) )
169, 10, 11, 12, 14letrd 9219 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  <_  D )
179, 11, 16abssuble0d 12227 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( B  -  D )
)  =  ( D  -  B ) )
1813, 15, 173brtr4d 4234 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  <_  ( abs `  ( B  -  D
) ) )
191, 3, 4, 6, 8, 18rlimsqzlem 12434 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   RRcr 8981    <_ cle 9113    - cmin 9283   abscabs 12031    ~~> r crli 12271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-rlim 12275
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