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Theorem bj-ccinftydisj 31725
Description: The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-ccinftydisj  |-  ( CC 
i^i CCinfty )  =  (/)

Proof of Theorem bj-ccinftydisj
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inftyexpidisj 31722 . . . 4  |-  -.  (inftyexpi  `  y )  e.  CC
21nex 1686 . . 3  |-  -.  E. y (inftyexpi  `  y )  e.  CC
3 elin 3608 . . . . . 6  |-  ( x  e.  ( CC  i^i CCinfty )  <-> 
( x  e.  CC  /\  x  e. CCinfty ) )
4 df-bj-inftyexpi 31719 . . . . . . . . . . 11  |- inftyexpi  =  ( z  e.  ( -u pi (,] pi )  |->  <.
z ,  CC >. )
54funmpt2 5626 . . . . . . . . . 10  |-  Fun inftyexpi
6 elrnrexdm 6041 . . . . . . . . . 10  |-  ( Fun inftyexpi  -> 
( x  e.  ran inftyexpi  ->  E. y  e.  dom inftyexpi  x  =  (inftyexpi  `  y ) ) )
75, 6ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ran inftyexpi  ->  E. y  e.  dom inftyexpi  x  =  (inftyexpi  `  y ) )
8 rexex 2843 . . . . . . . . 9  |-  ( E. y  e.  dom inftyexpi  x  =  (inftyexpi  `  y )  ->  E. y  x  =  (inftyexpi  `
 y ) )
97, 8syl 17 . . . . . . . 8  |-  ( x  e.  ran inftyexpi  ->  E. y  x  =  (inftyexpi  `  y
) )
10 df-bj-ccinfty 31724 . . . . . . . 8  |- CCinfty  =  ran inftyexpi
119, 10eleq2s 2567 . . . . . . 7  |-  ( x  e. CCinfty  ->  E. y  x  =  (inftyexpi  `  y ) )
1211anim2i 579 . . . . . 6  |-  ( ( x  e.  CC  /\  x  e. CCinfty )  ->  (
x  e.  CC  /\  E. y  x  =  (inftyexpi  `  y ) ) )
133, 12sylbi 200 . . . . 5  |-  ( x  e.  ( CC  i^i CCinfty )  ->  ( x  e.  CC  /\  E. y  x  =  (inftyexpi  `  y
) ) )
14 ancom 457 . . . . . 6  |-  ( ( x  e.  CC  /\  E. y  x  =  (inftyexpi  `  y ) )  <->  ( E. y  x  =  (inftyexpi  `  y )  /\  x  e.  CC ) )
15 exancom 1730 . . . . . . 7  |-  ( E. y ( x  e.  CC  /\  x  =  (inftyexpi  `  y ) )  <->  E. y ( x  =  (inftyexpi  `  y )  /\  x  e.  CC )
)
16 19.41v 1838 . . . . . . 7  |-  ( E. y ( x  =  (inftyexpi  `  y )  /\  x  e.  CC )  <->  ( E. y  x  =  (inftyexpi  `  y )  /\  x  e.  CC )
)
1715, 16bitri 257 . . . . . 6  |-  ( E. y ( x  e.  CC  /\  x  =  (inftyexpi  `  y ) )  <-> 
( E. y  x  =  (inftyexpi  `  y )  /\  x  e.  CC ) )
1814, 17sylbb2 221 . . . . 5  |-  ( ( x  e.  CC  /\  E. y  x  =  (inftyexpi  `  y ) )  ->  E. y ( x  e.  CC  /\  x  =  (inftyexpi  `  y ) ) )
1913, 18syl 17 . . . 4  |-  ( x  e.  ( CC  i^i CCinfty )  ->  E. y ( x  e.  CC  /\  x  =  (inftyexpi  `  y ) ) )
20 eleq1 2537 . . . . . 6  |-  ( x  =  (inftyexpi  `  y )  ->  ( x  e.  CC  <->  (inftyexpi  `  y )  e.  CC ) )
2120biimpac 494 . . . . 5  |-  ( ( x  e.  CC  /\  x  =  (inftyexpi  `  y
) )  ->  (inftyexpi  `  y )  e.  CC )
2221eximi 1715 . . . 4  |-  ( E. y ( x  e.  CC  /\  x  =  (inftyexpi  `  y ) )  ->  E. y (inftyexpi  `  y
)  e.  CC )
2319, 22syl 17 . . 3  |-  ( x  e.  ( CC  i^i CCinfty )  ->  E. y (inftyexpi  `  y
)  e.  CC )
242, 23mto 181 . 2  |-  -.  x  e.  ( CC  i^i CCinfty )
2524bj-nel0 31611 1  |-  ( CC 
i^i CCinfty )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   E.wrex 2757    i^i cin 3389   (/)c0 3722   <.cop 3965   dom cdm 4839   ran crn 4840   Fun wfun 5583   ` cfv 5589  (class class class)co 6308   CCcc 9555   -ucneg 9881   (,]cioc 11661   picpi 14196  inftyexpi cinftyexpi 31718  CCinftycccinfty 31723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-cnex 9613
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-c 9563  df-bj-inftyexpi 31719  df-bj-ccinfty 31724
This theorem is referenced by: (None)
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