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Theorem rlimsqz2 12403
Description: Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-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 )
rlimsqz2.1  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  B )
rlimsqz2.2  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  <_  C )
Assertion
Ref Expression
rlimsqz2  |-  ( 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 rlimsqz2
StepHypRef Expression
1 rlimsqz.m . 2  |-  ( ph  ->  M  e.  RR )
2 rlimsqz.d . . 3  |-  ( ph  ->  D  e.  RR )
32recnd 9074 . 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 9074 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
7 rlimsqz.c . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
87recnd 9074 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
97adantrr 698 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  e.  RR )
105adantrr 698 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  e.  RR )
112adantr 452 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  e.  RR )
12 rlimsqz2.1 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  B )
139, 10, 11, 12lesub1dd 9602 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( C  -  D
)  <_  ( B  -  D ) )
14 rlimsqz2.2 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  <_  C )
1511, 9, 14abssubge0d 12193 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  =  ( C  -  D ) )
1611, 9, 10, 14, 12letrd 9187 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  <_  B )
1711, 10, 16abssubge0d 12193 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( B  -  D )
)  =  ( B  -  D ) )
1813, 15, 173brtr4d 4206 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  <_  ( abs `  ( B  -  D
) ) )
191, 3, 4, 6, 8, 18rlimsqzlem 12401 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   class class class wbr 4176    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   RRcr 8949    <_ cle 9081    - cmin 9251   abscabs 11998    ~~> r crli 12238
This theorem is referenced by:  cxp2limlem  20771  cxp2lim  20772  chpchtlim  21130  selberg2lem  21201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-rlim 12242
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