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Theorem bj-inftyexpidisj 31652
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 4166 . . . . 5  |-  ( x  =  A  ->  <. x ,  CC >.  =  <. A ,  CC >. )
2 df-bj-inftyexpi 31649 . . . . 5  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
3 opex 4664 . . . . 5  |-  <. A ,  CC >.  e.  _V
41, 2, 3fvmpt 5948 . . . 4  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  =  <. A ,  CC >. )
5 opex 4664 . . . . 5  |-  <. x ,  CC >.  e.  _V
65, 2dmmpti 5707 . . . 4  |-  dom inftyexpi  =  (
-u pi (,] pi )
74, 6eleq2s 2547 . . 3  |-  ( A  e.  dom inftyexpi  ->  (inftyexpi  `  A
)  =  <. A ,  CC >. )
8 cnex 9620 . . . . . . 7  |-  CC  e.  _V
98prid2 4081 . . . . . 6  |-  CC  e.  { A ,  CC }
10 eqid 2451 . . . . . . . 8  |-  { A ,  CC }  =  { A ,  CC }
1110olci 393 . . . . . . 7  |-  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } )
12 elopg 4666 . . . . . . . 8  |-  ( ( A  e.  _V  /\  CC  e.  _V )  -> 
( { A ,  CC }  e.  <. A ,  CC >. 
<->  ( { A ,  CC }  =  { A }  \/  { A ,  CC }  =  { A ,  CC } ) ) )
138, 12mpan2 677 . . . . . . 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 8121 . . . . . . 7  |-  -.  ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >.  /\  <. A ,  CC >.  e.  CC )
1615bj-imn3ani 31171 . . . . . 6  |-  ( ( CC  e.  { A ,  CC }  /\  { A ,  CC }  e.  <. A ,  CC >. )  ->  -.  <. A ,  CC >.  e.  CC )
179, 14, 16sylancr 669 . . . . 5  |-  ( A  e.  _V  ->  -.  <. A ,  CC >.  e.  CC )
18 opprc1 4189 . . . . . 6  |-  ( -.  A  e.  _V  ->  <. A ,  CC >.  =  (/) )
19 0ncn 9557 . . . . . . 7  |-  -.  (/)  e.  CC
20 eleq1 2517 . . . . . . 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 2458 . . . . . 6  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  <->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2524biimpi 198 . . . . 5  |-  ( (inftyexpi  `  A )  =  <. A ,  CC >.  ->  <. A ,  CC >.  =  (inftyexpi  `  A
) )
2625eleq1d 2513 . . . 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 5889 . . . 4  |-  ( -.  A  e.  dom inftyexpi  ->  (inftyexpi  `  A )  =  (/) )
3029eleq1d 2513 . . 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 1444    e. wcel 1887   _Vcvv 3045   (/)c0 3731   {csn 3968   {cpr 3970   <.cop 3974   dom cdm 4834   ` cfv 5582  (class class class)co 6290   CCcc 9537   -ucneg 9861   (,]cioc 11636   picpi 14119  inftyexpi cinftyexpi 31648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-cnex 9595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-c 9545  df-bj-inftyexpi 31649
This theorem is referenced by:  bj-ccinftydisj  31655  bj-pinftynrr  31664  bj-minftynrr  31668
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