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Theorem rlimrecl 13057
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 9338 . . 3  |-  RR  C_  CC
43a1i 11 . 2  |-  ( ph  ->  RR  C_  CC )
5 eldifi 3477 . . . . . 6  |-  ( y  e.  ( CC  \  RR )  ->  y  e.  CC )
65adantl 466 . . . . 5  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  y  e.  CC )
76imcld 12683 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  RR )
87recnd 9411 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  CC )
9 eldifn 3478 . . . . 5  |-  ( y  e.  ( CC  \  RR )  ->  -.  y  e.  RR )
109adantl 466 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  -.  y  e.  RR )
11 reim0b 12607 . . . . . 6  |-  ( y  e.  CC  ->  (
y  e.  RR  <->  ( Im `  y )  =  0 ) )
126, 11syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( y  e.  RR  <->  ( Im `  y )  =  0 ) )
1312necon3bbid 2641 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( -.  y  e.  RR  <->  ( Im `  y )  =/=  0
) )
1410, 13mpbid 210 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  =/=  0
)
158, 14absrpcld 12933 . 2  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( abs `  ( Im `  y
) )  e.  RR+ )
166adantr 465 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  y  e.  CC )
17 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  RR )
1817recnd 9411 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  CC )
1916, 18subcld 9718 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
y  -  z )  e.  CC )
20 absimle 12797 . . . 4  |-  ( ( y  -  z )  e.  CC  ->  ( abs `  ( Im `  ( y  -  z
) ) )  <_ 
( abs `  (
y  -  z ) ) )
2119, 20syl 16 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  ( y  -  z
) ) )  <_ 
( abs `  (
y  -  z ) ) )
2216, 18imsubd 12705 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  ( y  -  z ) )  =  ( ( Im
`  y )  -  ( Im `  z ) ) )
23 reim0 12606 . . . . . . 7  |-  ( z  e.  RR  ->  (
Im `  z )  =  0 )
2423adantl 466 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  z )  =  0 )
2524oveq2d 6106 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  ( Im
`  z ) )  =  ( ( Im
`  y )  - 
0 ) )
268adantr 465 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  e.  CC )
2726subid1d 9707 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  0 )  =  ( Im `  y ) )
2822, 25, 273eqtrrd 2479 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  =  ( Im `  ( y  -  z
) ) )
2928fveq2d 5694 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  y ) )  =  ( abs `  (
Im `  ( y  -  z ) ) ) )
3018, 16abssubd 12938 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( z  -  y ) )  =  ( abs `  (
y  -  z ) ) )
3121, 29, 303brtr4d 4321 . 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 13055 1  |-  ( ph  ->  C  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605    \ cdif 3324    C_ wss 3327   class class class wbr 4291    e. cmpt 4349   ` cfv 5417  (class class class)co 6090   supcsup 7689   CCcc 9279   RRcr 9280   0cc0 9281   +oocpnf 9414   RR*cxr 9416    < clt 9417    <_ cle 9418    - cmin 9594   Imcim 12586   abscabs 12722    ~~> r crli 12962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-rlim 12966
This theorem is referenced by:  rlimge0  13058  climrecl  13060  rlimle  13124  divsqrsumo1  22376  mulog2sumlem1  22782
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