Theorem bj-inftyexpidisj 32891

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 4170	. . . . 5 |- ( x = A -> <. x , CC >. = <. A , CC >. )
2		df-bj-inftyexpi 32888	. . . . 5 |- inftyexpi = ( x e. ( -u pi (,] pi ) |-> <. x , CC >. )
3		opex 4667	. . . . 5 |- <. A , CC >. e. _V
4	1, 2, 3	fvmpt 5886	. . . 4 |- ( A e. ( -u pi (,] pi ) -> (inftyexpi ` A ) = <. A , CC >. )
5		opex 4667	. . . . 5 |- <. x , CC >. e. _V
6	5, 2	dmmpti 5651	. . . 4 |- dom inftyexpi = ( -u pi (,] pi )
7	4, 6	eleq2s 2562	. . 3 |- ( A e. dom inftyexpi -> (inftyexpi ` A ) = <. A , CC >. )
8		cnex 9477	. . . . . . 7 |- CC e. _V
9	8	prid2 4095	. . . . . 6 |- CC e. { A , CC }
10		eqid 2454	. . . . . . . 8 |- { A , CC } = { A , CC }
11	10	olci 391	. . . . . . 7 |- ( { A , CC } = { A } \/ { A , CC } = { A , CC } )
12		bj-elopg 32880	. . . . . . . 8 |- ( ( A e. _V /\ CC e. _V ) -> ( { A , CC } e. <. A , CC >. <-> ( { A , CC } = { A } \/ { A , CC } = { A , CC } ) ) )
13	8, 12	mpan2 671	. . . . . . 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 7936	. . . . . . 7 |- -. ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. /\ <. A , CC >. e. CC )
16	15	bj-imn3ani 32465	. . . . . 6 |- ( ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. ) -> -. <. A , CC >. e. CC )
17	9, 14, 16	sylancr 663	. . . . 5 |- ( A e. _V -> -. <. A , CC >. e. CC )
18		opprc1 4193	. . . . . 6 |- ( -. A e. _V -> <. A , CC >. = (/) )
19		0ncn 9414	. . . . . . 7 |- -. (/) e. CC
20		eleq1 2526	. . . . . . 7 |- ( <. A , CC >. = (/) -> ( <. A , CC >. e. CC <-> (/) e. CC ) )
21	19, 20	mtbiri 303	. . . . . 6 |- ( <. A , CC >. = (/) -> -. <. A , CC >. e. CC )
22	18, 21	syl 16	. . . . 5 |- ( -. A e. _V -> -. <. A , CC >. e. CC )
23	17, 22	pm2.61i 164	. . . 4 |- -. <. A , CC >. e. CC
24		eqcom 2463	. . . . . 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 2523	. . . 4 |- ( (inftyexpi ` A ) = <. A , CC >. -> ( <. A , CC >. e. CC <-> (inftyexpi ` A ) e. CC ) )
27	23, 26	mtbii 302	. . 3 |- ( (inftyexpi ` A ) = <. A , CC >. -> -. (inftyexpi ` A ) e. CC )
28	7, 27	syl 16	. 2 |- ( A e. dom inftyexpi -> -. (inftyexpi ` A ) e. CC )
29		ndmfv 5826	. . . 4 |- ( -. A e. dom inftyexpi -> (inftyexpi ` A ) = (/) )
30	29	eleq1d 2523	. . 3 |- ( -. A e. dom inftyexpi -> ( (inftyexpi ` A ) e. CC <-> (/) e. CC ) )
31	19, 30	mtbiri 303	. 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 368   = wceq 1370   e. wcel 1758   _Vcvv 3078   (/)c0 3748   {csn 3988   {cpr 3990   <.cop 3994   dom cdm 4951   ` cfv 5529  (class class class)co 6203   CCcc 9394   -ucneg 9710   (,]cioc 11415   picpi 13473  inftyexpi cinftyexpi 32887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-cnex 9452

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537  df-c 9402  df-bj-inftyexpi 32888

This theorem is referenced by:  bj-ccinftydisj  32894  bj-pinftynrr  32903  bj-minftynrr  32907