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Theorem bj-inftyexpidisj 31160
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 4156 . . . . 5  |-  ( x  =  A  ->  <. x ,  CC >.  =  <. A ,  CC >. )
2 df-bj-inftyexpi 31157 . . . . 5  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
3 opex 4652 . . . . 5  |-  <. A ,  CC >.  e.  _V
41, 2, 3fvmpt 5886 . . . 4  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  =  <. A ,  CC >. )
5 opex 4652 . . . . 5  |-  <. x ,  CC >.  e.  _V
65, 2dmmpti 5647 . . . 4  |-  dom inftyexpi  =  (
-u pi (,] pi )
74, 6eleq2s 2508 . . 3  |-  ( A  e.  dom inftyexpi  ->  (inftyexpi  `  A
)  =  <. A ,  CC >. )
8 cnex 9521 . . . . . . 7  |-  CC  e.  _V
98prid2 4078 . . . . . 6  |-  CC  e.  { A ,  CC }
10 eqid 2400 . . . . . . . 8  |-  { A ,  CC }  =  { A ,  CC }
1110olci 389 . . . . . . 7  |-  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } )
12 bj-elopg 31150 . . . . . . . 8  |-  ( ( A  e.  _V  /\  CC  e.  _V )  -> 
( { A ,  CC }  e.  <. A ,  CC >. 
<->  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } ) ) )
138, 12mpan2 669 . . . . . . 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 7984 . . . . . . 7  |-  -.  ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >.  /\  <. A ,  CC >.  e.  CC )
1615bj-imn3ani 30723 . . . . . 6  |-  ( ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >. )  ->  -.  <. A ,  CC >.  e.  CC )
179, 14, 16sylancr 661 . . . . 5  |-  ( A  e.  _V  ->  -.  <. A ,  CC >.  e.  CC )
18 opprc1 4179 . . . . . 6  |-  ( -.  A  e.  _V  ->  <. A ,  CC >.  =  (/) )
19 0ncn 9458 . . . . . . 7  |-  -.  (/)  e.  CC
20 eleq1 2472 . . . . . . 7  |-  ( <. A ,  CC >.  =  (/)  ->  ( <. A ,  CC >.  e.  CC  <->  (/)  e.  CC ) )
2119, 20mtbiri 301 . . . . . 6  |-  ( <. A ,  CC >.  =  (/)  ->  -.  <. A ,  CC >.  e.  CC )
2218, 21syl 17 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
<. A ,  CC >.  e.  CC )
2317, 22pm2.61i 164 . . . 4  |-  -.  <. A ,  CC >.  e.  CC
24 eqcom 2409 . . . . . 6  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  <->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2524biimpi 194 . . . . 5  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2625eleq1d 2469 . . . 4  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  ( <. A ,  CC >.  e.  CC  <->  (inftyexpi  `  A )  e.  CC ) )
2723, 26mtbii 300 . . 3  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  -.  (inftyexpi  `
 A )  e.  CC )
287, 27syl 17 . 2  |-  ( A  e.  dom inftyexpi  ->  -.  (inftyexpi  `  A )  e.  CC )
29 ndmfv 5827 . . . 4  |-  ( -.  A  e.  dom inftyexpi  ->  (inftyexpi  `  A )  =  (/) )
3029eleq1d 2469 . . 3  |-  ( -.  A  e.  dom inftyexpi  ->  (
(inftyexpi  `  A )  e.  CC  <->  (/)  e.  CC ) )
3119, 30mtbiri 301 . 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 366    = wceq 1403    e. wcel 1840   _Vcvv 3056   (/)c0 3735   {csn 3969   {cpr 3971   <.cop 3975   dom cdm 4940   ` cfv 5523  (class class class)co 6232   CCcc 9438   -ucneg 9760   (,]cioc 11499   picpi 13901  inftyexpi cinftyexpi 31156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-reg 7970  ax-cnex 9496
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531  df-c 9446  df-bj-inftyexpi 31157
This theorem is referenced by:  bj-ccinftydisj  31163  bj-pinftynrr  31172  bj-minftynrr  31176
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