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Theorem bj-inftyexpidisj 32891
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 4170 . . . . 5  |-  ( x  =  A  ->  <. x ,  CC >.  =  <. A ,  CC >. )
2 df-bj-inftyexpi 32888 . . . . 5  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
3 opex 4667 . . . . 5  |-  <. A ,  CC >.  e.  _V
41, 2, 3fvmpt 5886 . . . 4  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  =  <. A ,  CC >. )
5 opex 4667 . . . . 5  |-  <. x ,  CC >.  e.  _V
65, 2dmmpti 5651 . . . 4  |-  dom inftyexpi  =  (
-u pi (,] pi )
74, 6eleq2s 2562 . . 3  |-  ( A  e.  dom inftyexpi  ->  (inftyexpi  `  A
)  =  <. A ,  CC >. )
8 cnex 9477 . . . . . . 7  |-  CC  e.  _V
98prid2 4095 . . . . . 6  |-  CC  e.  { A ,  CC }
10 eqid 2454 . . . . . . . 8  |-  { A ,  CC }  =  { A ,  CC }
1110olci 391 . . . . . . 7  |-  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } )
12 bj-elopg 32880 . . . . . . . 8  |-  ( ( A  e.  _V  /\  CC  e.  _V )  -> 
( { A ,  CC }  e.  <. A ,  CC >. 
<->  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } ) ) )
138, 12mpan2 671 . . . . . . 7  |-  ( A  e.  _V  ->  ( { A ,  CC }  e.  <. A ,  CC >.  <-> 
( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } ) ) )
1411, 13mpbiri 233 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  CC }  e.  <. A ,  CC >. )
15 en3lp 7936 . . . . . . 7  |-  -.  ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >.  /\  <. A ,  CC >.  e.  CC )
1615bj-imn3ani 32465 . . . . . 6  |-  ( ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >. )  ->  -.  <. A ,  CC >.  e.  CC )
179, 14, 16sylancr 663 . . . . 5  |-  ( A  e.  _V  ->  -.  <. A ,  CC >.  e.  CC )
18 opprc1 4193 . . . . . 6  |-  ( -.  A  e.  _V  ->  <. A ,  CC >.  =  (/) )
19 0ncn 9414 . . . . . . 7  |-  -.  (/)  e.  CC
20 eleq1 2526 . . . . . . 7  |-  ( <. A ,  CC >.  =  (/)  ->  ( <. A ,  CC >.  e.  CC  <->  (/)  e.  CC ) )
2119, 20mtbiri 303 . . . . . 6  |-  ( <. A ,  CC >.  =  (/)  ->  -.  <. A ,  CC >.  e.  CC )
2218, 21syl 16 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
<. A ,  CC >.  e.  CC )
2317, 22pm2.61i 164 . . . 4  |-  -.  <. A ,  CC >.  e.  CC
24 eqcom 2463 . . . . . 6  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  <->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2524biimpi 194 . . . . 5  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2625eleq1d 2523 . . . 4  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  ( <. A ,  CC >.  e.  CC  <->  (inftyexpi  `  A )  e.  CC ) )
2723, 26mtbii 302 . . 3  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  -.  (inftyexpi  `
 A )  e.  CC )
287, 27syl 16 . 2  |-  ( A  e.  dom inftyexpi  ->  -.  (inftyexpi  `  A )  e.  CC )
29 ndmfv 5826 . . . 4  |-  ( -.  A  e.  dom inftyexpi  ->  (inftyexpi  `  A )  =  (/) )
3029eleq1d 2523 . . 3  |-  ( -.  A  e.  dom inftyexpi  ->  (
(inftyexpi  `  A )  e.  CC  <->  (/)  e.  CC ) )
3119, 30mtbiri 303 . 2  |-  ( -.  A  e.  dom inftyexpi  ->  -.  (inftyexpi  `
 A )  e.  CC )
3228, 31pm2.61i 164 1  |-  -.  (inftyexpi  `  A )  e.  CC
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3748   {csn 3988   {cpr 3990   <.cop 3994   dom cdm 4951   ` cfv 5529  (class class class)co 6203   CCcc 9394   -ucneg 9710   (,]cioc 11415   picpi 13473  inftyexpi cinftyexpi 32887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-cnex 9452
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537  df-c 9402  df-bj-inftyexpi 32888
This theorem is referenced by:  bj-ccinftydisj  32894  bj-pinftynrr  32903  bj-minftynrr  32907
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