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Theorem rlimconst 12018
Description: A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimconst  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  (
x  e.  A  |->  B )  ~~> r  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem rlimconst
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 8838 . . . 4  |-  0  e.  RR
2 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  B  e.  CC )
32subidd 9145 . . . . . . . . 9  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( B  -  B )  =  0 )
43fveq2d 5529 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( abs `  ( B  -  B
) )  =  ( abs `  0 ) )
5 abs0 11770 . . . . . . . 8  |-  ( abs `  0 )  =  0
64, 5syl6eq 2331 . . . . . . 7  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( abs `  ( B  -  B
) )  =  0 )
7 rpgt0 10365 . . . . . . . 8  |-  ( y  e.  RR+  ->  0  < 
y )
87ad2antlr 707 . . . . . . 7  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  0  <  y )
96, 8eqbrtrd 4043 . . . . . 6  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( abs `  ( B  -  B
) )  <  y
)
109a1d 22 . . . . 5  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( 0  <_  x  ->  ( abs `  ( B  -  B ) )  < 
y ) )
1110ralrimiva 2626 . . . 4  |-  ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  ->  A. x  e.  A  ( 0  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )
)
12 breq1 4026 . . . . . . 7  |-  ( z  =  0  ->  (
z  <_  x  <->  0  <_  x ) )
1312imbi1d 308 . . . . . 6  |-  ( z  =  0  ->  (
( z  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )  <->  ( 0  <_  x  ->  ( abs `  ( B  -  B ) )  <  y ) ) )
1413ralbidv 2563 . . . . 5  |-  ( z  =  0  ->  ( A. x  e.  A  ( z  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )  <->  A. x  e.  A  ( 0  <_  x  ->  ( abs `  ( B  -  B ) )  <  y ) ) )
1514rspcev 2884 . . . 4  |-  ( ( 0  e.  RR  /\  A. x  e.  A  ( 0  <_  x  ->  ( abs `  ( B  -  B ) )  <  y ) )  ->  E. z  e.  RR  A. x  e.  A  ( z  <_  x  ->  ( abs `  ( B  -  B ) )  <  y ) )
161, 11, 15sylancr 644 . . 3  |-  ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  ->  E. z  e.  RR  A. x  e.  A  ( z  <_  x  ->  ( abs `  ( B  -  B ) )  <  y ) )
1716ralrimiva 2626 . 2  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  A. y  e.  RR+  E. z  e.  RR  A. x  e.  A  ( z  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )
)
18 simplr 731 . . . 4  |-  ( ( ( A  C_  RR  /\  B  e.  CC )  /\  x  e.  A
)  ->  B  e.  CC )
1918ralrimiva 2626 . . 3  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  A. x  e.  A  B  e.  CC )
20 simpl 443 . . 3  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  A  C_  RR )
21 simpr 447 . . 3  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  B  e.  CC )
2219, 20, 21rlim2 11970 . 2  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  (
( x  e.  A  |->  B )  ~~> r  B  <->  A. y  e.  RR+  E. z  e.  RR  A. x  e.  A  ( z  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )
) )
2317, 22mpbird 223 1  |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  (
x  e.  A  |->  B )  ~~> r  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868    - cmin 9037   RR+crp 10354   abscabs 11719    ~~> r crli 11959
This theorem is referenced by:  o1const  12093  rlimneg  12120  caucvgr  12148  fsumrlim  12269  dvfsumrlimge0  19377  dvfsumrlim2  19379  logexprlim  20464  chebbnd2  20626  chto1lb  20627  chpchtlim  20628  dchrisum0lem1  20665  selberglem2  20695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-rlim 11963
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