MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimsqz2 Structured version   Unicode version

Theorem rlimsqz2 13558
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 9611 . 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 9611 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
7 rlimsqz.c . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
87recnd 9611 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
97adantrr 714 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  e.  RR )
105adantrr 714 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  e.  RR )
112adantr 463 . . . 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 10164 . . 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 13348 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  =  ( C  -  D ) )
1611, 9, 10, 14, 12letrd 9728 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  <_  B )
1711, 10, 16abssubge0d 13348 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( B  -  D )
)  =  ( B  -  D ) )
1813, 15, 173brtr4d 4469 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  M  <_  x ) )  -> 
( abs `  ( C  -  D )
)  <_  ( abs `  ( B  -  D
) ) )
191, 3, 4, 6, 8, 18rlimsqzlem 13556 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   RRcr 9480    <_ cle 9618    - cmin 9796   abscabs 13152    ~~> r crli 13393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-rlim 13397
This theorem is referenced by:  cxp2limlem  23506  cxp2lim  23507  chpchtlim  23865  selberg2lem  23936
  Copyright terms: Public domain W3C validator