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Theorem rlimsqz2 12214
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 8948 . 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 8948 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
7 rlimsqz.c . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
87recnd 8948 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
97adantrr 697 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  e.  RR )
105adantrr 697 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  e.  RR )
112adantr 451 . . . 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 9475 . . 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 12004 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  =  ( C  -  D ) )
1611, 9, 10, 14, 12letrd 9060 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  <_  B )
1711, 10, 16abssubge0d 12004 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( B  -  D )
)  =  ( B  -  D ) )
1813, 15, 173brtr4d 4132 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  <_  ( abs `  ( B  -  D
) ) )
191, 3, 4, 6, 8, 18rlimsqzlem 12212 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   class class class wbr 4102    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   RRcr 8823    <_ cle 8955    - cmin 9124   abscabs 11809    ~~> r crli 12049
This theorem is referenced by:  cxp2limlem  20375  cxp2lim  20376  chpchtlim  20734  selberg2lem  20805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-ico 10751  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-rlim 12053
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