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Theorem rlimconst 12034
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 8854 . . . 4  |-  0  e.  RR
2 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  B  e.  CC )
32subidd 9161 . . . . . . . . 9  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( B  -  B )  =  0 )
43fveq2d 5545 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( abs `  ( B  -  B
) )  =  ( abs `  0 ) )
5 abs0 11786 . . . . . . . 8  |-  ( abs `  0 )  =  0
64, 5syl6eq 2344 . . . . . . 7  |-  ( ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  /\  x  e.  A
)  ->  ( abs `  ( B  -  B
) )  =  0 )
7 rpgt0 10381 . . . . . . . 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 4059 . . . . . 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 2639 . . . 4  |-  ( ( ( A  C_  RR  /\  B  e.  CC )  /\  y  e.  RR+ )  ->  A. x  e.  A  ( 0  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )
)
12 breq1 4042 . . . . . . 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 2576 . . . . 5  |-  ( z  =  0  ->  ( A. x  e.  A  ( z  <_  x  ->  ( abs `  ( B  -  B )
)  <  y )  <->  A. x  e.  A  ( 0  <_  x  ->  ( abs `  ( B  -  B ) )  <  y ) ) )
1514rspcev 2897 . . . 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 2639 . 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 2639 . . 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 11986 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    < clt 8883    <_ cle 8884    - cmin 9053   RR+crp 10370   abscabs 11735    ~~> r crli 11975
This theorem is referenced by:  o1const  12109  rlimneg  12136  caucvgr  12164  fsumrlim  12285  dvfsumrlimge0  19393  dvfsumrlim2  19395  logexprlim  20480  chebbnd2  20642  chto1lb  20643  chpchtlim  20644  dchrisum0lem1  20681  selberglem2  20711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-rlim 11979
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