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Theorem rlimrecl 12376
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 9049 . . 3  |-  RR  C_  CC
43a1i 11 . 2  |-  ( ph  ->  RR  C_  CC )
5 eldifi 3471 . . . . . 6  |-  ( y  e.  ( CC  \  RR )  ->  y  e.  CC )
65adantl 454 . . . . 5  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  y  e.  CC )
76imcld 12002 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  RR )
87recnd 9116 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  e.  CC )
9 eldifn 3472 . . . . 5  |-  ( y  e.  ( CC  \  RR )  ->  -.  y  e.  RR )
109adantl 454 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  -.  y  e.  RR )
11 reim0b 11926 . . . . . 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 2637 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( -.  y  e.  RR  <->  ( Im `  y )  =/=  0
) )
1410, 13mpbid 203 . . 3  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( Im `  y )  =/=  0
)
158, 14absrpcld 12252 . 2  |-  ( (
ph  /\  y  e.  ( CC  \  RR ) )  ->  ( abs `  ( Im `  y
) )  e.  RR+ )
166adantr 453 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  y  e.  CC )
17 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  RR )
1817recnd 9116 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  z  e.  CC )
1916, 18subcld 9413 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
y  -  z )  e.  CC )
20 absimle 12116 . . . 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 12024 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  ( y  -  z ) )  =  ( ( Im
`  y )  -  ( Im `  z ) ) )
23 reim0 11925 . . . . . . 7  |-  ( z  e.  RR  ->  (
Im `  z )  =  0 )
2423adantl 454 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  z )  =  0 )
2524oveq2d 6099 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  ( Im
`  z ) )  =  ( ( Im
`  y )  - 
0 ) )
268adantr 453 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  e.  CC )
2726subid1d 9402 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
( Im `  y
)  -  0 )  =  ( Im `  y ) )
2822, 25, 273eqtrrd 2475 . . . 4  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  (
Im `  y )  =  ( Im `  ( y  -  z
) ) )
2928fveq2d 5734 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( Im `  y ) )  =  ( abs `  (
Im `  ( y  -  z ) ) ) )
3018, 16abssubd 12257 . . 3  |-  ( ( ( ph  /\  y  e.  ( CC  \  RR ) )  /\  z  e.  RR )  ->  ( abs `  ( z  -  y ) )  =  ( abs `  (
y  -  z ) ) )
3121, 29, 303brtr4d 4244 . 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 12374 1  |-  ( ph  ->  C  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   supcsup 7447   CCcc 8990   RRcr 8991   0cc0 8992    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   Imcim 11905   abscabs 12041    ~~> r crli 12281
This theorem is referenced by:  rlimge0  12377  climrecl  12379  rlimle  12443  divsqrsumo1  20824  mulog2sumlem1  21230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-rlim 12285
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