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.

Step	Hyp	Ref	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
4	1, 2, 3	fvmpt 5886	. . . 4 |- ( A e. ( -u pi (,] pi ) -> (inftyexpi ` A ) = <. A , CC >. )
5		opex 4652	. . . . 5 |- <. x , CC >. e. _V
6	5, 2	dmmpti 5647	. . . 4 |- dom inftyexpi = ( -u pi (,] pi )
7	4, 6	eleq2s 2508	. . 3 |- ( A e. dom inftyexpi -> (inftyexpi ` A ) = <. A , CC >. )
8		cnex 9521	. . . . . . 7 |- CC e. _V
9	8	prid2 4078	. . . . . 6 |- CC e. { A , CC }
10		eqid 2400	. . . . . . . 8 |- { A , CC } = { A , CC }
11	10	olci 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 } ) ) )
13	8, 12	mpan2 669	. . . . . . 7 |- ( A e. _V -> ( { A , CC } e. <. A , CC >. <-> ( { A , CC } = { A } \/ { A , CC } = { A , CC } ) ) )
14	11, 13	mpbiri 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 )
16	15	bj-imn3ani 30723	. . . . . 6 |- ( ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. ) -> -. <. A , CC >. e. CC )
17	9, 14, 16	sylancr 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 ) )
21	19, 20	mtbiri 301	. . . . . 6 |- ( <. A , CC >. = (/) -> -. <. A , CC >. e. CC )
22	18, 21	syl 17	. . . . 5 |- ( -. A e. _V -> -. <. A , CC >. e. CC )
23	17, 22	pm2.61i 164	. . . 4 |- -. <. A , CC >. e. CC
24		eqcom 2409	. . . . . 6 |- ( (inftyexpi ` A ) = <. A , CC >. <-> <. A , CC >. = (inftyexpi ` A ) )
25	24	biimpi 194	. . . . 5 |- ( (inftyexpi ` A ) = <. A , CC >. -> <. A , CC >. = (inftyexpi ` A ) )
26	25	eleq1d 2469	. . . 4 |- ( (inftyexpi ` A ) = <. A , CC >. -> ( <. A , CC >. e. CC <-> (inftyexpi ` A ) e. CC ) )
27	23, 26	mtbii 300	. . 3 |- ( (inftyexpi ` A ) = <. A , CC >. -> -. (inftyexpi ` A ) e. CC )
28	7, 27	syl 17	. 2 |- ( A e. dom inftyexpi -> -. (inftyexpi ` A ) e. CC )
29		ndmfv 5827	. . . 4 |- ( -. A e. dom inftyexpi -> (inftyexpi ` A ) = (/) )
30	29	eleq1d 2469	. . 3 |- ( -. A e. dom inftyexpi -> ( (inftyexpi ` A ) e. CC <-> (/) e. CC ) )
31	19, 30	mtbiri 301	. 2 |- ( -. A e. dom inftyexpi -> -. (inftyexpi ` A ) e. CC )
32	28, 31	pm2.61i 164	1 |- -. (inftyexpi ` A ) e. CC

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