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.

Step	Hyp	Ref	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
4	1, 2, 3	fvmpt 5962	. . . 4 |- ( A e. ( -u pi (,] pi ) -> (inftyexpi ` A ) = <. A , CC >. )
5		opex 4683	. . . . 5 |- <. x , CC >. e. _V
6	5, 2	dmmpti 5723	. . . 4 |- dom inftyexpi = ( -u pi (,] pi )
7	4, 6	eleq2s 2531	. . 3 |- ( A e. dom inftyexpi -> (inftyexpi ` A ) = <. A , CC >. )
8		cnex 9622	. . . . . . 7 |- CC e. _V
9	8	prid2 4107	. . . . . 6 |- CC e. { A , CC }
10		eqid 2423	. . . . . . . 8 |- { A , CC } = { A , CC }
11	10	olci 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 } ) ) )
13	8, 12	mpan2 676	. . . . . . 7 |- ( A e. _V -> ( { A , CC } e. <. A , CC >. <-> ( { A , CC } = { A } \/ { A , CC } = { A , CC } ) ) )
14	11, 13	mpbiri 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 )
16	15	bj-imn3ani 31169	. . . . . 6 |- ( ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. ) -> -. <. A , CC >. e. CC )
17	9, 14, 16	sylancr 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 ) )
21	19, 20	mtbiri 305	. . . . . 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 168	. . . 4 |- -. <. A , CC >. e. CC
24		eqcom 2432	. . . . . 6 |- ( (inftyexpi ` A ) = <. A , CC >. <-> <. A , CC >. = (inftyexpi ` A ) )
25	24	biimpi 198	. . . . 5 |- ( (inftyexpi ` A ) = <. A , CC >. -> <. A , CC >. = (inftyexpi ` A ) )
26	25	eleq1d 2492	. . . 4 |- ( (inftyexpi ` A ) = <. A , CC >. -> ( <. A , CC >. e. CC <-> (inftyexpi ` A ) e. CC ) )
27	23, 26	mtbii 304	. . 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 5903	. . . 4 |- ( -. A e. dom inftyexpi -> (inftyexpi ` A ) = (/) )
30	29	eleq1d 2492	. . 3 |- ( -. A e. dom inftyexpi -> ( (inftyexpi ` A ) e. CC <-> (/) e. CC ) )
31	19, 30	mtbiri 305	. 2 |- ( -. A e. dom inftyexpi -> -. (inftyexpi ` A ) e. CC )
32	28, 31	pm2.61i 168	1 |- -. (inftyexpi ` A ) e. CC

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