Theorem recex 10659

Description: Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)

Assertion

Ref	Expression
recex	|- ( ( A e. CC /\ A =/= 0 ) -> E. x e. CC ( A x. x ) = 1 )

Distinct variable group:   x, A

Proof of Theorem recex

Dummy variables y a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 10036	. . 3 |- ( A e. CC -> E. a e. RR E. b e. RR A = ( a + ( _i x. b ) ) )
2		recextlem2 10658	. . . . . . . . 9 |- ( ( a e. RR /\ b e. RR /\ ( a + ( _i x. b ) ) =/= 0 ) -> ( ( a x. a ) + ( b x. b ) ) =/= 0 )
3	2	3expia 1267	. . . . . . . 8 |- ( ( a e. RR /\ b e. RR ) -> ( ( a + ( _i x. b ) ) =/= 0 -> ( ( a x. a ) + ( b x. b ) ) =/= 0 ) )
4		remulcl 10021	. . . . . . . . . . . . 13 |- ( ( a e. RR /\ a e. RR ) -> ( a x. a ) e. RR )
5	4	anidms 677	. . . . . . . . . . . 12 |- ( a e. RR -> ( a x. a ) e. RR )
6		remulcl 10021	. . . . . . . . . . . . 13 |- ( ( b e. RR /\ b e. RR ) -> ( b x. b ) e. RR )
7	6	anidms 677	. . . . . . . . . . . 12 |- ( b e. RR -> ( b x. b ) e. RR )
8		readdcl 10019	. . . . . . . . . . . 12 |- ( ( ( a x. a ) e. RR /\ ( b x. b ) e. RR ) -> ( ( a x. a ) + ( b x. b ) ) e. RR )
9	5, 7, 8	syl2an 494	. . . . . . . . . . 11 |- ( ( a e. RR /\ b e. RR ) -> ( ( a x. a ) + ( b x. b ) ) e. RR )
10		ax-rrecex 10008	. . . . . . . . . . 11 |- ( ( ( ( a x. a ) + ( b x. b ) ) e. RR /\ ( ( a x. a ) + ( b x. b ) ) =/= 0 ) -> E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 )
11	9, 10	sylan 488	. . . . . . . . . 10 |- ( ( ( a e. RR /\ b e. RR ) /\ ( ( a x. a ) + ( b x. b ) ) =/= 0 ) -> E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 )
12		recn 10026	. . . . . . . . . . . 12 |- ( a e. RR -> a e. CC )
13		recn 10026	. . . . . . . . . . . 12 |- ( b e. RR -> b e. CC )
14		recn 10026	. . . . . . . . . . . . . 14 |- ( y e. RR -> y e. CC )
15		ax-icn 9995	. . . . . . . . . . . . . . . . . . . 20 |- _i e. CC
16		mulcl 10020	. . . . . . . . . . . . . . . . . . . 20 |- ( ( _i e. CC /\ b e. CC ) -> ( _i x. b ) e. CC )
17	15, 16	mpan 706	. . . . . . . . . . . . . . . . . . 19 |- ( b e. CC -> ( _i x. b ) e. CC )
18		subcl 10280	. . . . . . . . . . . . . . . . . . 19 |- ( ( a e. CC /\ ( _i x. b ) e. CC ) -> ( a - ( _i x. b ) ) e. CC )
19	17, 18	sylan2 491	. . . . . . . . . . . . . . . . . 18 |- ( ( a e. CC /\ b e. CC ) -> ( a - ( _i x. b ) ) e. CC )
20		mulcl 10020	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a - ( _i x. b ) ) e. CC /\ y e. CC ) -> ( ( a - ( _i x. b ) ) x. y ) e. CC )
21	19, 20	sylan 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( a - ( _i x. b ) ) x. y ) e. CC )
22	21	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) /\ ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 ) -> ( ( a - ( _i x. b ) ) x. y ) e. CC )
23		addcl 10018	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( a e. CC /\ ( _i x. b ) e. CC ) -> ( a + ( _i x. b ) ) e. CC )
24	17, 23	sylan2 491	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a e. CC /\ b e. CC ) -> ( a + ( _i x. b ) ) e. CC )
25	24	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( a + ( _i x. b ) ) e. CC )
26	19	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( a - ( _i x. b ) ) e. CC )
27		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> y e. CC )
28	25, 26, 27	mulassd 10063	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) x. y ) = ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) )
29		recextlem1 10657	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a e. CC /\ b e. CC ) -> ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) = ( ( a x. a ) + ( b x. b ) ) )
30	29	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) = ( ( a x. a ) + ( b x. b ) ) )
31	30	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) x. y ) = ( ( ( a x. a ) + ( b x. b ) ) x. y ) )
32	28, 31	eqtr3d 2658	. . . . . . . . . . . . . . . . 17 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = ( ( ( a x. a ) + ( b x. b ) ) x. y ) )
33		id 22	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 )
34	32, 33	sylan9eq 2676	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) /\ ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 ) -> ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = 1 )
35		oveq2 6658	. . . . . . . . . . . . . . . . . 18 |- ( x = ( ( a - ( _i x. b ) ) x. y ) -> ( ( a + ( _i x. b ) ) x. x ) = ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) )
36	35	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( x = ( ( a - ( _i x. b ) ) x. y ) -> ( ( ( a + ( _i x. b ) ) x. x ) = 1 <-> ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = 1 ) )
37	36	rspcev 3309	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a - ( _i x. b ) ) x. y ) e. CC /\ ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = 1 ) -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 )
38	22, 34, 37	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) /\ ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 ) -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 )
39	38	exp31 630	. . . . . . . . . . . . . 14 |- ( ( a e. CC /\ b e. CC ) -> ( y e. CC -> ( ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) ) )
40	14, 39	syl5 34	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( y e. RR -> ( ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) ) )
41	40	rexlimdv 3030	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
42	12, 13, 41	syl2an 494	. . . . . . . . . . 11 |- ( ( a e. RR /\ b e. RR ) -> ( E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
43	42	adantr 481	. . . . . . . . . 10 |- ( ( ( a e. RR /\ b e. RR ) /\ ( ( a x. a ) + ( b x. b ) ) =/= 0 ) -> ( E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
44	11, 43	mpd 15	. . . . . . . . 9 |- ( ( ( a e. RR /\ b e. RR ) /\ ( ( a x. a ) + ( b x. b ) ) =/= 0 ) -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 )
45	44	ex 450	. . . . . . . 8 |- ( ( a e. RR /\ b e. RR ) -> ( ( ( a x. a ) + ( b x. b ) ) =/= 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
46	3, 45	syld 47	. . . . . . 7 |- ( ( a e. RR /\ b e. RR ) -> ( ( a + ( _i x. b ) ) =/= 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
47	46	adantr 481	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( ( a + ( _i x. b ) ) =/= 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
48		neeq1 2856	. . . . . . 7 |- ( A = ( a + ( _i x. b ) ) -> ( A =/= 0 <-> ( a + ( _i x. b ) ) =/= 0 ) )
49	48	adantl 482	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( A =/= 0 <-> ( a + ( _i x. b ) ) =/= 0 ) )
50		oveq1 6657	. . . . . . . . 9 |- ( A = ( a + ( _i x. b ) ) -> ( A x. x ) = ( ( a + ( _i x. b ) ) x. x ) )
51	50	eqeq1d 2624	. . . . . . . 8 |- ( A = ( a + ( _i x. b ) ) -> ( ( A x. x ) = 1 <-> ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
52	51	rexbidv 3052	. . . . . . 7 |- ( A = ( a + ( _i x. b ) ) -> ( E. x e. CC ( A x. x ) = 1 <-> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
53	52	adantl 482	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( E. x e. CC ( A x. x ) = 1 <-> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
54	47, 49, 53	3imtr4d 283	. . . . 5 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( A =/= 0 -> E. x e. CC ( A x. x ) = 1 ) )
55	54	ex 450	. . . 4 |- ( ( a e. RR /\ b e. RR ) -> ( A = ( a + ( _i x. b ) ) -> ( A =/= 0 -> E. x e. CC ( A x. x ) = 1 ) ) )
56	55	rexlimivv 3036	. . 3 |- ( E. a e. RR E. b e. RR A = ( a + ( _i x. b ) ) -> ( A =/= 0 -> E. x e. CC ( A x. x ) = 1 ) )
57	1, 56	syl 17	. 2 |- ( A e. CC -> ( A =/= 0 -> E. x e. CC ( A x. x ) = 1 ) )
58	57	imp 445	1 |- ( ( A e. CC /\ A =/= 0 ) -> E. x e. CC ( A x. x ) = 1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266

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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  mulcand  10660  receu  10672