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Theorem rlimrecl 12054
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 8794 . . 3  |-  RR  C_  CC
43a1i 10 . 2  |-  ( ph  ->  RR  C_  CC )
5 eldifi 3298 . . . . . 6  |-  ( y  e.  ( CC  \  RR )  ->  y  e.  CC )
65adantl 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  y  e.  CC )
76imcld 11680 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  RR )
87recnd 8861 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  CC )
9 eldifn 3299 . . . . 5  |-  ( y  e.  ( CC  \  RR )  ->  -.  y  e.  RR )
109adantl 452 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  -.  y  e.  RR )
11 reim0b 11604 . . . . . 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 2480 . . . 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 11930 . 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 8861 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  CC )
1916, 18subcld 9157 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
y  -  z )  e.  CC )
20 absimle 11794 . . . 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 11702 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  ( y  -  z ) )  =  ( ( Im
`  y )  -  ( Im `  z ) ) )
23 reim0 11603 . . . . . . 7  |-  ( z  e.  RR  ->  (
Im `  z )  =  0 )
2423adantl 452 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  z )  =  0 )
2524oveq2d 5874 . . . . 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 9146 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  0 )  =  ( Im `  y ) )
2822, 25, 273eqtrrd 2320 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  =  ( Im `  ( y  -  z
) ) )
2928fveq2d 5529 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  y ) )  =  ( abs `  (
Im `  ( y  -  z ) ) ) )
3018, 16abssubd 11935 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( z  -  y ) )  =  ( abs `  (
y  -  z ) ) )
3121, 29, 303brtr4d 4053 . 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 12052 1  |-  ( ph  ->  C  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   Imcim 11583   abscabs 11719    ~~> r crli 11959
This theorem is referenced by:  rlimge0  12055  climrecl  12057  rlimle  12121  divsqrsumo1  20278  mulog2sumlem1  20683
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  ax-pre-sup 8815
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-sup 7194  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-3 9805  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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