Theorem cnegex 6502

Description: Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)

Assertion

Ref	Expression
cnegex	|- (A e. CC -> E.x e. CC (A + x) = 0)

Distinct variable group:   x,A

Proof of Theorem cnegex

Step	Hyp	Ref	Expression
1		axcnre 6439	. 2 |- (A e. CC -> E.a e. RR E.b e. RR A = (a + (_i x. b)))
2		opreq1 4889	. . . . . 6 |- (A = (a + (_i x. b)) -> (A + x) = ((a + (_i x. b)) + x))
3	2	eqeq1d 1892	. . . . 5 |- (A = (a + (_i x. b)) -> ((A + x) = 0 <-> ((a + (_i x. b)) + x) = 0))
4	3	rexbidv 2124	. . . 4 |- (A = (a + (_i x. b)) -> (E.x e. CC (A + x) = 0 <-> E.x e. CC ((a + (_i x. b)) + x) = 0))
5		cnegexlem2 6500	. . . . 5 |- E.y e. RR (_i x. y) e. RR
6		cnegexlem3 6501	. . . . . . . . 9 |- ((b e. RR /\ y e. RR) -> E.c e. RR (b + c) = y)
7	6	ad2ant2lr 446	. . . . . . . 8 |- (((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) -> E.c e. RR (b + c) = y)
8		readdcl 6455	. . . . . . . . . . . . . 14 |- ((a e. RR /\ (_i x. y) e. RR) -> (a + (_i x. y)) e. RR)
9		axrnegex 6436	. . . . . . . . . . . . . 14 |- ((a + (_i x. y)) e. RR -> E.d e. RR ((a + (_i x. y)) + d) = 0)
10	8, 9	syl 12	. . . . . . . . . . . . 13 |- ((a e. RR /\ (_i x. y) e. RR) -> E.d e. RR ((a + (_i x. y)) + d) = 0)
11	10	ad2ant2rl 447	. . . . . . . . . . . 12 |- (((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) -> E.d e. RR ((a + (_i x. y)) + d) = 0)
12	11	adantr 425	. . . . . . . . . . 11 |- ((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) -> E.d e. RR ((a + (_i x. y)) + d) = 0)
13		addcl 6454	. . . . . . . . . . . . . . . . . 18 |- (((_i x. c) e. CC /\ d e. CC) -> ((_i x. c) + d) e. CC)
14		mulcl 6456	. . . . . . . . . . . . . . . . . . 19 |- ((_i e. CC /\ c e. CC) -> (_i x. c) e. CC)
15		axicn 6423	. . . . . . . . . . . . . . . . . . 19 |- _i e. CC
16		recn 6466	. . . . . . . . . . . . . . . . . . 19 |- (c e. RR -> c e. CC)
17	14, 15, 16	sylancr 526	. . . . . . . . . . . . . . . . . 18 |- (c e. RR -> (_i x. c) e. CC)
18		recn 6466	. . . . . . . . . . . . . . . . . 18 |- (d e. RR -> d e. CC)
19	13, 17, 18	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((c e. RR /\ d e. RR) -> ((_i x. c) + d) e. CC)
20	19	adantlr 429	. . . . . . . . . . . . . . . 16 |- (((c e. RR /\ (b + c) = y) /\ d e. RR) -> ((_i x. c) + d) e. CC)
21	20	adantll 428	. . . . . . . . . . . . . . 15 |- (((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) -> ((_i x. c) + d) e. CC)
22	21	adantr 425	. . . . . . . . . . . . . 14 |- ((((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) /\ ((a + (_i x. y)) + d) = 0) -> ((_i x. c) + d) e. CC)
23		addcl 6454	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((a e. CC /\ (_i x. b) e. CC) -> (a + (_i x. b)) e. CC)
24		mulcl 6456	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((_i e. CC /\ b e. CC) -> (_i x. b) e. CC)
25	15, 24	mpan 759	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (b e. CC -> (_i x. b) e. CC)
26	23, 25	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((a e. CC /\ b e. CC) -> (a + (_i x. b)) e. CC)
27	26	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> (a + (_i x. b)) e. CC)
28	15, 14	mpan 759	. . . . . . . . . . . . . . . . . . . . . . 23 |- (c e. CC -> (_i x. c) e. CC)
29	28	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> (_i x. c) e. CC)
30		simpr 350	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> d e. CC)
31		addass 6460	. . . . . . . . . . . . . . . . . . . . . 22 |- (((a + (_i x. b)) e. CC /\ (_i x. c) e. CC /\ d e. CC) -> (((a + (_i x. b)) + (_i x. c)) + d) = ((a + (_i x. b)) + ((_i x. c) + d)))
32	27, 29, 30, 31	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . 21 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> (((a + (_i x. b)) + (_i x. c)) + d) = ((a + (_i x. b)) + ((_i x. c) + d)))
33		simpll 448	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> a e. CC)
34	25	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> (_i x. b) e. CC)
35	28	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> (_i x. c) e. CC)
36		addass 6460	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((a e. CC /\ (_i x. b) e. CC /\ (_i x. c) e. CC) -> ((a + (_i x. b)) + (_i x. c)) = (a + ((_i x. b) + (_i x. c))))
37	33, 34, 35, 36	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> ((a + (_i x. b)) + (_i x. c)) = (a + ((_i x. b) + (_i x. c))))
38		adddi 6462	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((_i e. CC /\ b e. CC /\ c e. CC) -> (_i x. (b + c)) = ((_i x. b) + (_i x. c)))
39	15, 38	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((b e. CC /\ c e. CC) -> (_i x. (b + c)) = ((_i x. b) + (_i x. c)))
40	39	adantll 428	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> (_i x. (b + c)) = ((_i x. b) + (_i x. c)))
41	40	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> (a + (_i x. (b + c))) = (a + ((_i x. b) + (_i x. c))))
42	37, 41	eqtr4d 1928	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((a e. CC /\ b e. CC) /\ c e. CC) -> ((a + (_i x. b)) + (_i x. c)) = (a + (_i x. (b + c))))
43	42	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> ((a + (_i x. b)) + (_i x. c)) = (a + (_i x. (b + c))))
44	43	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . 21 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> (((a + (_i x. b)) + (_i x. c)) + d) = ((a + (_i x. (b + c))) + d))
45	32, 44	eqtr3d 1927	. . . . . . . . . . . . . . . . . . . 20 |- ((((a e. CC /\ b e. CC) /\ c e. CC) /\ d e. CC) -> ((a + (_i x. b)) + ((_i x. c) + d)) = ((a + (_i x. (b + c))) + d))
46		recn 6466	. . . . . . . . . . . . . . . . . . . . . 22 |- (a e. RR -> a e. CC)
47		recn 6466	. . . . . . . . . . . . . . . . . . . . . 22 |- (b e. RR -> b e. CC)
48	46, 47	anim12i 360	. . . . . . . . . . . . . . . . . . . . 21 |- ((a e. RR /\ b e. RR) -> (a e. CC /\ b e. CC))
49	48, 16	anim12i 360	. . . . . . . . . . . . . . . . . . . 20 |- (((a e. RR /\ b e. RR) /\ c e. RR) -> ((a e. CC /\ b e. CC) /\ c e. CC))
50	45, 49, 18	syl2an 503	. . . . . . . . . . . . . . . . . . 19 |- ((((a e. RR /\ b e. RR) /\ c e. RR) /\ d e. RR) -> ((a + (_i x. b)) + ((_i x. c) + d)) = ((a + (_i x. (b + c))) + d))
51	50	adantlrr 435	. . . . . . . . . . . . . . . . . 18 |- ((((a e. RR /\ b e. RR) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) -> ((a + (_i x. b)) + ((_i x. c) + d)) = ((a + (_i x. (b + c))) + d))
52		opreq2 4890	. . . . . . . . . . . . . . . . . . . . . 22 |- ((b + c) = y -> (_i x. (b + c)) = (_i x. y))
53	52	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . 21 |- ((b + c) = y -> (a + (_i x. (b + c))) = (a + (_i x. y)))
54	53	opreq1d 4897	. . . . . . . . . . . . . . . . . . . 20 |- ((b + c) = y -> ((a + (_i x. (b + c))) + d) = ((a + (_i x. y)) + d))
55	54	adantl 424	. . . . . . . . . . . . . . . . . . 19 |- ((c e. RR /\ (b + c) = y) -> ((a + (_i x. (b + c))) + d) = ((a + (_i x. y)) + d))
56	55	ad2antlr 441	. . . . . . . . . . . . . . . . . 18 |- ((((a e. RR /\ b e. RR) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) -> ((a + (_i x. (b + c))) + d) = ((a + (_i x. y)) + d))
57	51, 56	eqtr2d 1926	. . . . . . . . . . . . . . . . 17 |- ((((a e. RR /\ b e. RR) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) -> ((a + (_i x. y)) + d) = ((a + (_i x. b)) + ((_i x. c) + d)))
58	57	adantllr 433	. . . . . . . . . . . . . . . 16 |- (((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) -> ((a + (_i x. y)) + d) = ((a + (_i x. b)) + ((_i x. c) + d)))
59	58	eqeq1d 1892	. . . . . . . . . . . . . . 15 |- (((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) -> (((a + (_i x. y)) + d) = 0 <-> ((a + (_i x. b)) + ((_i x. c) + d)) = 0))
60	59	biimpa 460	. . . . . . . . . . . . . 14 |- ((((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) /\ ((a + (_i x. y)) + d) = 0) -> ((a + (_i x. b)) + ((_i x. c) + d)) = 0)
61		opreq2 4890	. . . . . . . . . . . . . . . 16 |- (x = ((_i x. c) + d) -> ((a + (_i x. b)) + x) = ((a + (_i x. b)) + ((_i x. c) + d)))
62	61	eqeq1d 1892	. . . . . . . . . . . . . . 15 |- (x = ((_i x. c) + d) -> (((a + (_i x. b)) + x) = 0 <-> ((a + (_i x. b)) + ((_i x. c) + d)) = 0))
63	62	rcla4ev 2381	. . . . . . . . . . . . . 14 |- ((((_i x. c) + d) e. CC /\ ((a + (_i x. b)) + ((_i x. c) + d)) = 0) -> E.x e. CC ((a + (_i x. b)) + x) = 0)
64	22, 60, 63	syl11anc 524	. . . . . . . . . . . . 13 |- ((((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) /\ d e. RR) /\ ((a + (_i x. y)) + d) = 0) -> E.x e. CC ((a + (_i x. b)) + x) = 0)
65	64	exp31 407	. . . . . . . . . . . 12 |- ((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) -> (d e. RR -> (((a + (_i x. y)) + d) = 0 -> E.x e. CC ((a + (_i x. b)) + x) = 0)))
66	65	r19.23adv 2215	. . . . . . . . . . 11 |- ((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) -> (E.d e. RR ((a + (_i x. y)) + d) = 0 -> E.x e. CC ((a + (_i x. b)) + x) = 0))
67	12, 66	mpd 29	. . . . . . . . . 10 |- ((((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) /\ (c e. RR /\ (b + c) = y)) -> E.x e. CC ((a + (_i x. b)) + x) = 0)
68	67	exp32 408	. . . . . . . . 9 |- (((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) -> (c e. RR -> ((b + c) = y -> E.x e. CC ((a + (_i x. b)) + x) = 0)))
69	68	r19.23adv 2215	. . . . . . . 8 |- (((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) -> (E.c e. RR (b + c) = y -> E.x e. CC ((a + (_i x. b)) + x) = 0))
70	7, 69	mpd 29	. . . . . . 7 |- (((a e. RR /\ b e. RR) /\ (y e. RR /\ (_i x. y) e. RR)) -> E.x e. CC ((a + (_i x. b)) + x) = 0)
71	70	exp32 408	. . . . . 6 |- ((a e. RR /\ b e. RR) -> (y e. RR -> ((_i x. y) e. RR -> E.x e. CC ((a + (_i x. b)) + x) = 0)))
72	71	r19.23adv 2215	. . . . 5 |- ((a e. RR /\ b e. RR) -> (E.y e. RR (_i x. y) e. RR -> E.x e. CC ((a + (_i x. b)) + x) = 0))
73	5, 72	mpi 55	. . . 4 |- ((a e. RR /\ b e. RR) -> E.x e. CC ((a + (_i x. b)) + x) = 0)
74	4, 73	syl5cbir 228	. . 3 |- ((a e. RR /\ b e. RR) -> (A = (a + (_i x. b)) -> E.x e. CC (A + x) = 0))
75	74	r19.23aivv 2217	. 2 |- (E.a e. RR E.b e. RR A = (a + (_i x. b)) -> E.x e. CC (A + x) = 0)
76	1, 75	syl 12	1 |- (A e. CC -> E.x e. CC (A + x) = 0)

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

This theorem is referenced by:  cnegexi 6503  0cnALT 6504

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-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-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  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-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

Copyright terms: Public domain