Theorem bj-inftyexpidisj 33097

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 4402	. . . . 5 |- ( x = A -> <. x , CC >. = <. A , CC >. )
2		df-bj-inftyexpi 33094	. . . . 5 |- inftyexpi = ( x e. ( -u pi (,] pi ) |-> <. x , CC >. )
3		opex 4932	. . . . 5 |- <. A , CC >. e. _V
4	1, 2, 3	fvmpt 6282	. . . 4 |- ( A e. ( -u pi (,] pi ) -> (inftyexpi ` A ) = <. A , CC >. )
5		opex 4932	. . . . 5 |- <. x , CC >. e. _V
6	5, 2	dmmpti 6023	. . . 4 |- dom inftyexpi = ( -u pi (,] pi )
7	4, 6	eleq2s 2719	. . 3 |- ( A e. dom inftyexpi -> (inftyexpi ` A ) = <. A , CC >. )
8		cnex 10017	. . . . . . 7 |- CC e. _V
9	8	prid2 4298	. . . . . 6 |- CC e. { A , CC }
10		eqid 2622	. . . . . . . 8 |- { A , CC } = { A , CC }
11	10	olci 406	. . . . . . 7 |- ( { A , CC } = { A } \/ { A , CC } = { A , CC } )
12		elopg 4934	. . . . . . . 8 |- ( ( A e. _V /\ CC e. _V ) -> ( { A , CC } e. <. A , CC >. <-> ( { A , CC } = { A } \/ { A , CC } = { A , CC } ) ) )
13	8, 12	mpan2 707	. . . . . . 7 |- ( A e. _V -> ( { A , CC } e. <. A , CC >. <-> ( { A , CC } = { A } \/ { A , CC } = { A , CC } ) ) )
14	11, 13	mpbiri 248	. . . . . 6 |- ( A e. _V -> { A , CC } e. <. A , CC >. )
15		en3lp 8513	. . . . . . 7 |- -. ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. /\ <. A , CC >. e. CC )
16	15	bj-imn3ani 32572	. . . . . 6 |- ( ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. ) -> -. <. A , CC >. e. CC )
17	9, 14, 16	sylancr 695	. . . . 5 |- ( A e. _V -> -. <. A , CC >. e. CC )
18		opprc1 4425	. . . . . 6 |- ( -. A e. _V -> <. A , CC >. = (/) )
19		0ncn 9954	. . . . . . 7 |- -. (/) e. CC
20		eleq1 2689	. . . . . . 7 |- ( <. A , CC >. = (/) -> ( <. A , CC >. e. CC <-> (/) e. CC ) )
21	19, 20	mtbiri 317	. . . . . 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 176	. . . 4 |- -. <. A , CC >. e. CC
24		eqcom 2629	. . . . . 6 |- ( (inftyexpi ` A ) = <. A , CC >. <-> <. A , CC >. = (inftyexpi ` A ) )
25	24	biimpi 206	. . . . 5 |- ( (inftyexpi ` A ) = <. A , CC >. -> <. A , CC >. = (inftyexpi ` A ) )
26	25	eleq1d 2686	. . . 4 |- ( (inftyexpi ` A ) = <. A , CC >. -> ( <. A , CC >. e. CC <-> (inftyexpi ` A ) e. CC ) )
27	23, 26	mtbii 316	. . 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 6218	. . . 4 |- ( -. A e. dom inftyexpi -> (inftyexpi ` A ) = (/) )
30	29	eleq1d 2686	. . 3 |- ( -. A e. dom inftyexpi -> ( (inftyexpi ` A ) e. CC <-> (/) e. CC ) )
31	19, 30	mtbiri 317	. 2 |- ( -. A e. dom inftyexpi -> -. (inftyexpi ` A ) e. CC )
32	28, 31	pm2.61i 176	1 |- -. (inftyexpi ` A ) e. CC

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   -ucneg 10267   (,]cioc 12176   picpi 14797  inftyexpi cinftyexpi 33093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-c 9942  df-bj-inftyexpi 33094

This theorem is referenced by:  bj-ccinftydisj  33100  bj-pinftynrr  33109  bj-minftynrr  33113