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Theorem bj-elccinfty 33982
Description: A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
Assertion
Ref Expression
bj-elccinfty  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  e. CCinfty )

Proof of Theorem bj-elccinfty
StepHypRef Expression
1 df-bj-inftyexpi 33975 . . . . 5  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
21funmpt2 5630 . . . 4  |-  Fun inftyexpi
32jctl 541 . . 3  |-  ( A  e.  dom inftyexpi  ->  ( Fun inftyexpi  /\  A  e.  dom inftyexpi  ) )
4 opex 4716 . . . . 5  |-  <. x ,  CC >.  e.  _V
54, 1dmmpti 5715 . . . 4  |-  dom inftyexpi  =  (
-u pi (,] pi )
65eqcomi 2480 . . 3  |-  ( -u pi (,] pi )  =  dom inftyexpi
73, 6eleq2s 2575 . 2  |-  ( A  e.  ( -u pi (,] pi )  ->  ( Fun inftyexpi 
/\  A  e.  dom inftyexpi  ) )
8 fvelrn 6024 . 2  |-  ( ( Fun inftyexpi  /\  A  e.  dom inftyexpi  )  ->  (inftyexpi  `  A )  e.  ran inftyexpi  )
9 df-bj-ccinfty 33980 . . . . 5  |- CCinfty  =  ran inftyexpi
109eqcomi 2480 . . . 4  |-  ran inftyexpi  = CCinfty
1110eleq2i 2545 . . 3  |-  ( (inftyexpi  `  A )  e.  ran inftyexpi  <->  (inftyexpi  `  A
)  e. CCinfty )
1211biimpi 194 . 2  |-  ( (inftyexpi  `  A )  e.  ran inftyexpi  -> 
(inftyexpi  `  A )  e. CCinfty )
137, 8, 123syl 20 1  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  e. CCinfty )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   <.cop 4038   dom cdm 5004   ran crn 5005   Fun wfun 5587   ` cfv 5593  (class class class)co 6294   CCcc 9500   -ucneg 9816   (,]cioc 11540   picpi 13676  inftyexpi cinftyexpi 33974  CCinftycccinfty 33979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-bj-inftyexpi 33975  df-bj-ccinfty 33980
This theorem is referenced by:  bj-pinftyccb  33989
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