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Theorem bj-elccinfty 31401
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 31394 . . . . 5  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
21funmpt2 5638 . . . 4  |-  Fun inftyexpi
32jctl 543 . . 3  |-  ( A  e.  dom inftyexpi  ->  ( Fun inftyexpi  /\  A  e.  dom inftyexpi  ) )
4 opex 4686 . . . . 5  |-  <. x ,  CC >.  e.  _V
54, 1dmmpti 5725 . . . 4  |-  dom inftyexpi  =  (
-u pi (,] pi )
65eqcomi 2442 . . 3  |-  ( -u pi (,] pi )  =  dom inftyexpi
73, 6eleq2s 2537 . 2  |-  ( A  e.  ( -u pi (,] pi )  ->  ( Fun inftyexpi 
/\  A  e.  dom inftyexpi  ) )
8 fvelrn 6030 . 2  |-  ( ( Fun inftyexpi  /\  A  e.  dom inftyexpi  )  ->  (inftyexpi  `  A )  e.  ran inftyexpi  )
9 df-bj-ccinfty 31399 . . . . 5  |- CCinfty  =  ran inftyexpi
109eqcomi 2442 . . . 4  |-  ran inftyexpi  = CCinfty
1110eleq2i 2507 . . 3  |-  ( (inftyexpi  `  A )  e.  ran inftyexpi  <->  (inftyexpi  `  A
)  e. CCinfty )
1211biimpi 197 . 2  |-  ( (inftyexpi  `  A )  e.  ran inftyexpi  -> 
(inftyexpi  `  A )  e. CCinfty )
137, 8, 123syl 18 1  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  e. CCinfty )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870   <.cop 4008   dom cdm 4854   ran crn 4855   Fun wfun 5595   ` cfv 5601  (class class class)co 6305   CCcc 9536   -ucneg 9860   (,]cioc 11636   picpi 14097  inftyexpi cinftyexpi 31393  CCinftycccinfty 31398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-bj-inftyexpi 31394  df-bj-ccinfty 31399
This theorem is referenced by:  bj-pinftyccb  31408
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