Theorem recex 5840

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 5440	. . 3 |- (A e. CC -> E.a e. RR E.b e. RR A = (a + (i x. b)))
2		recextlem2 5839	. . . . . . . . 9 |- ((a e. RR /\ b e. RR /\ (a + (i x. b)) =/= 0) -> ((a x. a) + (b x. b)) =/= 0)
3	2	3expia 841	. . . . . . . 8 |- ((a e. RR /\ b e. RR) -> ((a + (i x. b)) =/= 0 -> ((a x. a) + (b x. b)) =/= 0))
4		axrrecex 5438	. . . . . . . . . . 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 5456	. . . . . . . . . . . 12 |- (((a x. a) e. RR /\ (b x. b) e. RR) -> ((a x. a) + (b x. b)) e. RR)
6		remulcl 5458	. . . . . . . . . . . . 13 |- ((a e. RR /\ a e. RR) -> (a x. a) e. RR)
7	6	anidms 435	. . . . . . . . . . . 12 |- (a e. RR -> (a x. a) e. RR)
8		remulcl 5458	. . . . . . . . . . . . 13 |- ((b e. RR /\ b e. RR) -> (b x. b) e. RR)
9	8	anidms 435	. . . . . . . . . . . 12 |- (b e. RR -> (b x. b) e. RR)
10	5, 7, 9	syl2an 456	. . . . . . . . . . 11 |- ((a e. RR /\ b e. RR) -> ((a x. a) + (b x. b)) e. RR)
11	4, 10	sylan 450	. . . . . . . . . 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		opreq2 4027	. . . . . . . . . . . . . . . . . 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)))
13	12	eqeq1d 1526	. . . . . . . . . . . . . . . . 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))
14	13	rcla4ev 1923	. . . . . . . . . . . . . . . 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)
15		mulcl 5457	. . . . . . . . . . . . . . . . . 18 |- (((a - (i x. b)) e. CC /\ y e. CC) -> ((a - (i x. b)) x. y) e. CC)
16		subcl 5521	. . . . . . . . . . . . . . . . . . 19 |- ((a e. CC /\ (i x. b) e. CC) -> (a - (i x. b)) e. CC)
17		axicn 5424	. . . . . . . . . . . . . . . . . . . 20 |- i e. CC
18		mulcl 5457	. . . . . . . . . . . . . . . . . . . 20 |- ((i e. CC /\ b e. CC) -> (i x. b) e. CC)
19	17, 18	mpan 699	. . . . . . . . . . . . . . . . . . 19 |- (b e. CC -> (i x. b) e. CC)
20	16, 19	sylan2 453	. . . . . . . . . . . . . . . . . 18 |- ((a e. CC /\ b e. CC) -> (a - (i x. b)) e. CC)
21	15, 20	sylan 450	. . . . . . . . . . . . . . . . 17 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> ((a - (i x. b)) x. y) e. CC)
22	21	adantr 389	. . . . . . . . . . . . . . . 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		mulass 5462	. . . . . . . . . . . . . . . . . . 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)))
24		addcl 5455	. . . . . . . . . . . . . . . . . . . . 21 |- ((a e. CC /\ (i x. b) e. CC) -> (a + (i x. b)) e. CC)
25	24, 19	sylan2 453	. . . . . . . . . . . . . . . . . . . 20 |- ((a e. CC /\ b e. CC) -> (a + (i x. b)) e. CC)
26	25	adantr 389	. . . . . . . . . . . . . . . . . . 19 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> (a + (i x. b)) e. CC)
27	20	adantr 389	. . . . . . . . . . . . . . . . . . 19 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> (a - (i x. b)) e. CC)
28		pm3.27 321	. . . . . . . . . . . . . . . . . . 19 |- (((a e. CC /\ b e. CC) /\ y e. CC) -> y e. CC)
29	23, 26, 27, 28	syl3anc 864	. . . . . . . . . . . . . . . . . 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)))
30		recextlem1 5838	. . . . . . . . . . . . . . . . . . . 20 |- ((a e. CC /\ b e. CC) -> ((a + (i x. b)) x. (a - (i x. b))) = ((a x. a) + (b x. b)))
31	30	adantr 389	. . . . . . . . . . . . . . . . . . 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)))
32	31	opreq1d 4033	. . . . . . . . . . . . . . . . . 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))
33	29, 32	eqtr3d 1552	. . . . . . . . . . . . . . . . 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))
34		id 59	. . . . . . . . . . . . . . . . 17 |- ((((a x. a) + (b x. b)) x. y) = 1 -> (((a x. a) + (b x. b)) x. y) = 1)
35	33, 34	sylan9eq 1570	. . . . . . . . . . . . . . . 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)
36	14, 22, 35	sylanc 473	. . . . . . . . . . . . . . 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 376	. . . . . . . . . . . . . 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 5467	. . . . . . . . . . . . . 14 |- (y e. RR -> y e. CC)
39	37, 38	syl5 21	. . . . . . . . . . . . 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 1792	. . . . . . . . . . . 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 5467	. . . . . . . . . . . 12 |- (a e. RR -> a e. CC)
42		recn 5467	. . . . . . . . . . . 12 |- (b e. RR -> b e. CC)
43	40, 41, 42	syl2an 456	. . . . . . . . . . 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 389	. . . . . . . . . 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 26	. . . . . . . . 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 371	. . . . . . . 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 27	. . . . . . 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 389	. . . . . 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 1633	. . . . . . 7 |- (A = (a + (i x. b)) -> (A =/= 0 <-> (a + (i x. b)) =/= 0))
50	49	adantl 388	. . . . . 6 |- (((a e. RR /\ b e. RR) /\ A = (a + (i x. b))) -> (A =/= 0 <-> (a + (i x. b)) =/= 0))
51		opreq1 4026	. . . . . . . . 9 |- (A = (a + (i x. b)) -> (A x. x) = ((a + (i x. b)) x. x))
52	51	eqeq1d 1526	. . . . . . . 8 |- (A = (a + (i x. b)) -> ((A x. x) = 1 <-> ((a + (i x. b)) x. x) = 1))
53	52	rexbidv 1710	. . . . . . 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 388	. . . . . 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 546	. . . . 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 371	. . . 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 1794	. . 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 10	. 2 |- (A e. CC -> (A =/= 0 -> E.x e. CC (A x. x) = 1))
59	58	imp 348	1 |- ((A e. CC /\ A =/= 0) -> E.x e. CC (A x. x) = 1)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628  E.wrex 1692  (class class class)co 4021  CCcc 5386  RRcr 5387  0cc0 5388  1c1 5389  ici 5390   + caddc 5391   x. cmul 5393   - cmin 5446

This theorem is referenced by:  recexi 5841

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645

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