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Theorem bj-inftyexpidisj 31610
Description: An element of the circle at infinity is not a complex number. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
Assertion
Ref Expression
bj-inftyexpidisj  |-  -.  (inftyexpi  `  A )  e.  CC

Proof of Theorem bj-inftyexpidisj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opeq1 4185 . . . . 5  |-  ( x  =  A  ->  <. x ,  CC >.  =  <. A ,  CC >. )
2 df-bj-inftyexpi 31607 . . . . 5  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
3 opex 4683 . . . . 5  |-  <. A ,  CC >.  e.  _V
41, 2, 3fvmpt 5962 . . . 4  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  =  <. A ,  CC >. )
5 opex 4683 . . . . 5  |-  <. x ,  CC >.  e.  _V
65, 2dmmpti 5723 . . . 4  |-  dom inftyexpi  =  (
-u pi (,] pi )
74, 6eleq2s 2531 . . 3  |-  ( A  e.  dom inftyexpi  ->  (inftyexpi  `  A
)  =  <. A ,  CC >. )
8 cnex 9622 . . . . . . 7  |-  CC  e.  _V
98prid2 4107 . . . . . 6  |-  CC  e.  { A ,  CC }
10 eqid 2423 . . . . . . . 8  |-  { A ,  CC }  =  { A ,  CC }
1110olci 393 . . . . . . 7  |-  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } )
12 bj-elopg 31600 . . . . . . . 8  |-  ( ( A  e.  _V  /\  CC  e.  _V )  -> 
( { A ,  CC }  e.  <. A ,  CC >. 
<->  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } ) ) )
138, 12mpan2 676 . . . . . . 7  |-  ( A  e.  _V  ->  ( { A ,  CC }  e.  <. A ,  CC >.  <-> 
( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } ) ) )
1411, 13mpbiri 237 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  CC }  e.  <. A ,  CC >. )
15 en3lp 8125 . . . . . . 7  |-  -.  ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >.  /\  <. A ,  CC >.  e.  CC )
1615bj-imn3ani 31169 . . . . . 6  |-  ( ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >. )  ->  -.  <. A ,  CC >.  e.  CC )
179, 14, 16sylancr 668 . . . . 5  |-  ( A  e.  _V  ->  -.  <. A ,  CC >.  e.  CC )
18 opprc1 4208 . . . . . 6  |-  ( -.  A  e.  _V  ->  <. A ,  CC >.  =  (/) )
19 0ncn 9559 . . . . . . 7  |-  -.  (/)  e.  CC
20 eleq1 2495 . . . . . . 7  |-  ( <. A ,  CC >.  =  (/)  ->  ( <. A ,  CC >.  e.  CC  <->  (/)  e.  CC ) )
2119, 20mtbiri 305 . . . . . 6  |-  ( <. A ,  CC >.  =  (/)  ->  -.  <. A ,  CC >.  e.  CC )
2218, 21syl 17 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
<. A ,  CC >.  e.  CC )
2317, 22pm2.61i 168 . . . 4  |-  -.  <. A ,  CC >.  e.  CC
24 eqcom 2432 . . . . . 6  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  <->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2524biimpi 198 . . . . 5  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2625eleq1d 2492 . . . 4  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  ( <. A ,  CC >.  e.  CC  <->  (inftyexpi  `  A )  e.  CC ) )
2723, 26mtbii 304 . . 3  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  -.  (inftyexpi  `
 A )  e.  CC )
287, 27syl 17 . 2  |-  ( A  e.  dom inftyexpi  ->  -.  (inftyexpi  `  A )  e.  CC )
29 ndmfv 5903 . . . 4  |-  ( -.  A  e.  dom inftyexpi  ->  (inftyexpi  `  A )  =  (/) )
3029eleq1d 2492 . . 3  |-  ( -.  A  e.  dom inftyexpi  ->  (
(inftyexpi  `  A )  e.  CC  <->  (/)  e.  CC ) )
3119, 30mtbiri 305 . 2  |-  ( -.  A  e.  dom inftyexpi  ->  -.  (inftyexpi  `
 A )  e.  CC )
3228, 31pm2.61i 168 1  |-  -.  (inftyexpi  `  A )  e.  CC
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/ wo 370    = wceq 1438    e. wcel 1869   _Vcvv 3082   (/)c0 3762   {csn 3997   {cpr 3999   <.cop 4003   dom cdm 4851   ` cfv 5599  (class class class)co 6303   CCcc 9539   -ucneg 9863   (,]cioc 11638   picpi 14112  inftyexpi cinftyexpi 31606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-reg 8111  ax-cnex 9597
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fn 5602  df-fv 5607  df-c 9547  df-bj-inftyexpi 31607
This theorem is referenced by:  bj-ccinftydisj  31613  bj-pinftynrr  31622  bj-minftynrr  31626
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