Theorem recex 6876

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

Step	Hyp	Ref	Expression
1		axcnre 6439	. . 3 |- (A e. CC -> E.a e. RR E.b e. RR A = (a + (_i x. b)))
2		recextlem2 6875	. . . . . . . . 9 |- ((a e. RR /\ b e. RR /\ (a + (_i x. b)) =/= 0) -> ((a x. a) + (b x. b)) =/= 0)
3	2	3expia 1069	. . . . . . . 8 |- ((a e. RR /\ b e. RR) -> ((a + (_i x. b)) =/= 0 -> ((a x. a) + (b x. b)) =/= 0))
4		axrrecex 6437	. . . . . . . . . . 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)
5		readdcl 6455	. . . . . . . . . . . 12 |- (((a x. a) e. RR /\ (b x. b) e. RR) -> ((a x. a) + (b x. b)) e. RR)
6		remulcl 6457	. . . . . . . . . . . . 13 |- ((a e. RR /\ a e. RR) -> (a x. a) e. RR)
7	6	anidms 480	. . . . . . . . . . . 12 |- (a e. RR -> (a x. a) e. RR)
8		remulcl 6457	. . . . . . . . . . . . 13 |- ((b e. RR /\ b e. RR) -> (b x. b) e. RR)
9	8	anidms 480	. . . . . . . . . . . 12 |- (b e. RR -> (b x. b) e. RR)
10	5, 7, 9	syl2an 503	. . . . . . . . . . 11 |- ((a e. RR /\ b e. RR) -> ((a x. a) + (b x. b)) e. RR)
11	4, 10	sylan 497	. . . . . . . . . 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		mulcl 6456	. . . . . . . . . . . . . . . . . 18 |- (((a - (_i x. b)) e. CC /\ y e. CC) -> ((a - (_i x. b)) x. y) e. CC)
13		subcl 6524	. . . . . . . . . . . . . . . . . . 19 |- ((a e. CC /\ (_i x. b) e. CC) -> (a - (_i x. b)) e. CC)
14		axicn 6423	. . . . . . . . . . . . . . . . . . . 20 |- _i e. CC
15		mulcl 6456	. . . . . . . . . . . . . . . . . . . 20 |- ((_i e. CC /\ b e. CC) -> (_i x. b) e. CC)
16	14, 15	mpan 759	. . . . . . . . . . . . . . . . . . 19 |- (b e. CC -> (_i x. b) e. CC)
17	13, 16	sylan2 500	. . . . . . . . . . . . . . . . . 18 |- ((a e. CC /\ b e. CC) -> (a - (_i x. b)) e. CC)
18	12, 17	sylan 497	. . . . . . . . . . . . . . . . 17 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> ((a - (_i x. b)) x. y) e. CC)
19	18	adantr 425	. . . . . . . . . . . . . . . 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)
20		addcl 6454	. . . . . . . . . . . . . . . . . . . . 21 |- ((a e. CC /\ (_i x. b) e. CC) -> (a + (_i x. b)) e. CC)
21	20, 16	sylan2 500	. . . . . . . . . . . . . . . . . . . 20 |- ((a e. CC /\ b e. CC) -> (a + (_i x. b)) e. CC)
22	21	adantr 425	. . . . . . . . . . . . . . . . . . 19 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> (a + (_i x. b)) e. CC)
23	17	adantr 425	. . . . . . . . . . . . . . . . . . 19 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> (a - (_i x. b)) e. CC)
24		simpr 350	. . . . . . . . . . . . . . . . . . 19 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> y e. CC)
25		mulass 6461	. . . . . . . . . . . . . . . . . . 19 |- (((a + (_i x. b)) e. CC /\ (a - (_i x. 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)))
26	22, 23, 24, 25	syl111anc 1100	. . . . . . . . . . . . . . . . . 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)))
27		recextlem1 6874	. . . . . . . . . . . . . . . . . . . 20 |- ((a e. CC /\ b e. CC) -> ((a + (_i x. b)) x. (a - (_i x. b))) = ((a x. a) + (b x. b)))
28	27	adantr 425	. . . . . . . . . . . . . . . . . . 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)))
29	28	opreq1d 4897	. . . . . . . . . . . . . . . . . 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))
30	26, 29	eqtr3d 1927	. . . . . . . . . . . . . . . . 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))
31		id 73	. . . . . . . . . . . . . . . . 17 |- ((((a x. a) + (b x. b)) x. y) = 1 -> (((a x. a) + (b x. b)) x. y) = 1)
32	30, 31	sylan9eq 1948	. . . . . . . . . . . . . . . 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)
33		opreq2 4890	. . . . . . . . . . . . . . . . . 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)))
34	33	eqeq1d 1892	. . . . . . . . . . . . . . . . 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))
35	34	rcla4ev 2381	. . . . . . . . . . . . . . . 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)
36	19, 32, 35	syl11anc 524	. . . . . . . . . . . . . . 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)
37	36	exp31 407	. . . . . . . . . . . . . 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)))
38		recn 6466	. . . . . . . . . . . . . 14 |- (y e. RR -> y e. CC)
39	37, 38	syl5 20	. . . . . . . . . . . . 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)))
40	39	r19.23adv 2215	. . . . . . . . . . . 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))
41		recn 6466	. . . . . . . . . . . 12 |- (a e. RR -> a e. CC)
42		recn 6466	. . . . . . . . . . . 12 |- (b e. RR -> b e. CC)
43	40, 41, 42	syl2an 503	. . . . . . . . . . 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))
44	43	adantr 425	. . . . . . . . . 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))
45	11, 44	mpd 29	. . . . . . . . 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)
46	45	ex 402	. . . . . . . 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))
47	3, 46	syld 30	. . . . . . 7 |- ((a e. RR /\ b e. RR) -> ((a + (_i x. b)) =/= 0 -> E.x e. CC ((a + (_i x. b)) x. x) = 1))
48	47	adantr 425	. . . . . 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))
49		neeq1 2024	. . . . . . 7 |- (A = (a + (_i x. b)) -> (A =/= 0 <-> (a + (_i x. b)) =/= 0))
50	49	adantl 424	. . . . . 6 |- (((a e. RR /\ b e. RR) /\ A = (a + (_i x. b))) -> (A =/= 0 <-> (a + (_i x. b)) =/= 0))
51		opreq1 4889	. . . . . . . . 9 |- (A = (a + (_i x. b)) -> (A x. x) = ((a + (_i x. b)) x. x))
52	51	eqeq1d 1892	. . . . . . . 8 |- (A = (a + (_i x. b)) -> ((A x. x) = 1 <-> ((a + (_i x. b)) x. x) = 1))
53	52	rexbidv 2124	. . . . . . 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))
54	53	adantl 424	. . . . . 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))
55	48, 50, 54	3imtr4d 602	. . . . 5 |- (((a e. RR /\ b e. RR) /\ A = (a + (_i x. b))) -> (A =/= 0 -> E.x e. CC (A x. x) = 1))
56	55	ex 402	. . . 4 |- ((a e. RR /\ b e. RR) -> (A = (a + (_i x. b)) -> (A =/= 0 -> E.x e. CC (A x. x) = 1)))
57	56	r19.23aivv 2217	. . 3 |- (E.a e. RR E.b e. RR A = (a + (_i x. b)) -> (A =/= 0 -> E.x e. CC (A x. x) = 1))
58	1, 57	syl 12	. 2 |- (A e. CC -> (A =/= 0 -> E.x e. CC (A x. x) = 1))
59	58	imp 377	1 |- ((A e. CC /\ A =/= 0) -> E.x e. CC (A x. x) = 1)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445

This theorem is referenced by:  recexi 6877

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658

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