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Theorem rlimrecl 12070
Description: The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)
Hypotheses
Ref Expression
rlimcld2.1  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
rlimcld2.2  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  C
)
rlimrecl.3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Assertion
Ref Expression
rlimrecl  |-  ( ph  ->  C  e.  RR )
Distinct variable groups:    x, A    x, C    ph, x
Allowed substitution hint:    B( x)

Proof of Theorem rlimrecl
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimcld2.1 . 2  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
2 rlimcld2.2 . 2  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  C
)
3 ax-resscn 8810 . . 3  |-  RR  C_  CC
43a1i 10 . 2  |-  ( ph  ->  RR  C_  CC )
5 eldifi 3311 . . . . . 6  |-  ( y  e.  ( CC  \  RR )  ->  y  e.  CC )
65adantl 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  y  e.  CC )
76imcld 11696 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  RR )
87recnd 8877 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  CC )
9 eldifn 3312 . . . . 5  |-  ( y  e.  ( CC  \  RR )  ->  -.  y  e.  RR )
109adantl 452 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  -.  y  e.  RR )
11 reim0b 11620 . . . . . 6  |-  ( y  e.  CC  ->  (
y  e.  RR  <->  ( Im `  y )  =  0 ) )
126, 11syl 15 . . . . 5  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( y  e.  RR  <->  ( Im `  y )  =  0 ) )
1312necon3bbid 2493 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( -.  y  e.  RR  <->  ( Im `  y )  =/=  0
) )
1410, 13mpbid 201 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  =/=  0
)
158, 14absrpcld 11946 . 2  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( abs `  ( Im `  y
) )  e.  RR+ )
166adantr 451 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  y  e.  CC )
17 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  RR )
1817recnd 8877 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  CC )
1916, 18subcld 9173 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
y  -  z )  e.  CC )
20 absimle 11810 . . . 4  |-  ( ( y  -  z )  e.  CC  ->  ( abs `  ( Im `  ( y  -  z
) ) )  <_ 
( abs `  (
y  -  z ) ) )
2119, 20syl 15 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  ( y  -  z
) ) )  <_ 
( abs `  (
y  -  z ) ) )
2216, 18imsubd 11718 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  ( y  -  z ) )  =  ( ( Im
`  y )  -  ( Im `  z ) ) )
23 reim0 11619 . . . . . . 7  |-  ( z  e.  RR  ->  (
Im `  z )  =  0 )
2423adantl 452 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  z )  =  0 )
2524oveq2d 5890 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  ( Im
`  z ) )  =  ( ( Im
`  y )  - 
0 ) )
268adantr 451 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  e.  CC )
2726subid1d 9162 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  0 )  =  ( Im `  y ) )
2822, 25, 273eqtrrd 2333 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  =  ( Im `  ( y  -  z
) ) )
2928fveq2d 5545 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  y ) )  =  ( abs `  (
Im `  ( y  -  z ) ) ) )
3018, 16abssubd 11951 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( z  -  y ) )  =  ( abs `  (
y  -  z ) ) )
3121, 29, 303brtr4d 4069 . 2  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  y ) )  <_ 
( abs `  (
z  -  y ) ) )
32 rlimrecl.3 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
331, 2, 4, 15, 31, 32rlimcld2 12068 1  |-  ( ph  ->  C  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   Imcim 11599   abscabs 11735    ~~> r crli 11975
This theorem is referenced by:  rlimge0  12071  climrecl  12073  rlimle  12137  divsqrsumo1  20294  mulog2sumlem1  20699
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  ax-pre-sup 8831
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-sup 7210  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-3 9821  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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