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.

Step	Hyp	Ref	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
4	1, 2, 3	fvmpt 5948	. . . 4 |- ( A e. ( -u pi (,] pi ) -> (inftyexpi ` A ) = <. A , CC >. )
5		opex 4664	. . . . 5 |- <. x , CC >. e. _V
6	5, 2	dmmpti 5707	. . . 4 |- dom inftyexpi = ( -u pi (,] pi )
7	4, 6	eleq2s 2547	. . 3 |- ( A e. dom inftyexpi -> (inftyexpi ` A ) = <. A , CC >. )
8		cnex 9620	. . . . . . 7 |- CC e. _V
9	8	prid2 4081	. . . . . 6 |- CC e. { A , CC }
10		eqid 2451	. . . . . . . 8 |- { A , CC } = { A , CC }
11	10	olci 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 } ) ) )
13	8, 12	mpan2 677	. . . . . . 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 8121	. . . . . . 7 |- -. ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. /\ <. A , CC >. e. CC )
16	15	bj-imn3ani 31171	. . . . . 6 |- ( ( CC e. { A , CC } /\ { A , CC } e. <. A , CC >. ) -> -. <. A , CC >. e. CC )
17	9, 14, 16	sylancr 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 ) )
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 2458	. . . . . 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 2513	. . . 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 5889	. . . 4 |- ( -. A e. dom inftyexpi -> (inftyexpi ` A ) = (/) )
30	29	eleq1d 2513	. . 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 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