Theorem abs1mi 8156

Description: For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195.

Hypothesis

Ref	Expression
abs1m.1	|- A e. CC

Assertion

Ref	Expression
abs1mi	|- E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A))

Distinct variable group:   x,A

Proof of Theorem abs1mi

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . 6 |- (x = 1 -> (abs` x) = (abs` 1))
2	1	eqeq1d 1892	. . . . 5 |- (x = 1 -> ((abs` x) = 1 <-> (abs` 1) = 1))
3		opreq1 4889	. . . . . 6 |- (x = 1 -> (x x. A) = (1 x. A))
4	3	eqeq2d 1895	. . . . 5 |- (x = 1 -> ((abs` A) = (x x. A) <-> (abs` A) = (1 x. A)))
5	2, 4	anbi12d 690	. . . 4 |- (x = 1 -> (((abs` x) = 1 /\ (abs` A) = (x x. A)) <-> ((abs` 1) = 1 /\ (abs` A) = (1 x. A))))
6	5	rcla4ev 2381	. . 3 |- ((1 e. CC /\ ((abs` 1) = 1 /\ (abs` A) = (1 x. A))) -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
7		ax1cn 6422	. . 3 |- 1 e. CC
8		abs1m.1	. . . . . . 7 |- A e. CC
9	8	abs00i 8093	. . . . . 6 |- ((abs` A) = 0 <-> A = 0)
10	9	biimpri 169	. . . . 5 |- (A = 0 -> (abs` A) = 0)
11		opreq2 4890	. . . . . 6 |- (A = 0 -> (1 x. A) = (1 x. 0))
12		0cn 6481	. . . . . . 7 |- 0 e. CC
13	12	mulid2i 6486	. . . . . 6 |- (1 x. 0) = 0
14	11, 13	syl6eq 1944	. . . . 5 |- (A = 0 -> (1 x. A) = 0)
15	10, 14	eqtr4d 1928	. . . 4 |- (A = 0 -> (abs` A) = (1 x. A))
16		0re 6603	. . . . . 6 |- 0 e. RR
17		1re 6598	. . . . . 6 |- 1 e. RR
18		lt01 6871	. . . . . 6 |- 0 < 1
19	16, 17, 18	ltleii 6756	. . . . 5 |- 0 <_ 1
20	17	absidi 8112	. . . . 5 |- (0 <_ 1 -> (abs` 1) = 1)
21	19, 20	ax-mp 7	. . . 4 |- (abs` 1) = 1
22	15, 21	jctil 316	. . 3 |- (A = 0 -> ((abs` 1) = 1 /\ (abs` A) = (1 x. A)))
23	6, 7, 22	sylancr 526	. 2 |- (A = 0 -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
24	9	necon3bii 2032	. . . 4 |- ((abs` A) =/= 0 <-> A =/= 0)
25	8	cjcli 8017	. . . . . 6 |- (*` A) e. CC
26	8	abscli 8090	. . . . . . 7 |- (abs` A) e. RR
27	26	recni 6467	. . . . . 6 |- (abs` A) e. CC
28	25, 27	divclzi 6900	. . . . 5 |- ((abs` A) =/= 0 -> ((*` A) / (abs` A)) e. CC)
29	25, 27	absdivzi 8110	. . . . . 6 |- ((abs` A) =/= 0 -> (abs` ((*` A) / (abs` A))) = ((abs` (*` A)) / (abs` (abs` A))))
30		divid 6942	. . . . . . . 8 |- (((abs` A) e. CC /\ (abs` A) =/= 0) -> ((abs` A) / (abs` A)) = 1)
31	27, 30	mpan 759	. . . . . . 7 |- ((abs` A) =/= 0 -> ((abs` A) / (abs` A)) = 1)
32	8	abscji 8096	. . . . . . . 8 |- (abs` (*` A)) = (abs` A)
33		absidm 8144	. . . . . . . . 9 |- (A e. CC -> (abs` (abs` A)) = (abs` A))
34	8, 33	ax-mp 7	. . . . . . . 8 |- (abs` (abs` A)) = (abs` A)
35	32, 34	opreq12i 4894	. . . . . . 7 |- ((abs` (*` A)) / (abs` (abs` A))) = ((abs` A) / (abs` A))
36	31, 35	syl5eq 1940	. . . . . 6 |- ((abs` A) =/= 0 -> ((abs` (*` A)) / (abs` (abs` A))) = 1)
37	29, 36	eqtrd 1925	. . . . 5 |- ((abs` A) =/= 0 -> (abs` ((*` A) / (abs` A))) = 1)
38	8	absvalsqi 8088	. . . . . . . 8 |- ((abs` A)^2) = (A x. (*` A))
39	27	sqvali 7859	. . . . . . . 8 |- ((abs` A)^2) = ((abs` A) x. (abs` A))
40	8, 25	mulcomi 6476	. . . . . . . 8 |- (A x. (*` A)) = ((*` A) x. A)
41	38, 39, 40	3eqtr3i 1918	. . . . . . 7 |- ((abs` A) x. (abs` A)) = ((*` A) x. A)
42	25, 8	mulcli 6474	. . . . . . . 8 |- ((*` A) x. A) e. CC
43	42, 27, 27	divmulzi 6895	. . . . . . 7 |- ((abs` A) =/= 0 -> ((((*` A) x. A) / (abs` A)) = (abs` A) <-> ((abs` A) x. (abs` A)) = ((*` A) x. A)))
44	41, 43	mpbiri 211	. . . . . 6 |- ((abs` A) =/= 0 -> (((*` A) x. A) / (abs` A)) = (abs` A))
45		div23 6925	. . . . . . . 8 |- (((*` A) e. CC /\ A e. CC /\ ((abs` A) e. CC /\ (abs` A) =/= 0)) -> (((*` A) x. A) / (abs` A)) = (((*` A) / (abs` A)) x. A))
46	25, 8, 45	mp3an12 1181	. . . . . . 7 |- (((abs` A) e. CC /\ (abs` A) =/= 0) -> (((*` A) x. A) / (abs` A)) = (((*` A) / (abs` A)) x. A))
47	27, 46	mpan 759	. . . . . 6 |- ((abs` A) =/= 0 -> (((*` A) x. A) / (abs` A)) = (((*` A) / (abs` A)) x. A))
48	44, 47	eqtr3d 1927	. . . . 5 |- ((abs` A) =/= 0 -> (abs` A) = (((*` A) / (abs` A)) x. A))
49	28, 37, 48	jca32 312	. . . 4 |- ((abs` A) =/= 0 -> (((*` A) / (abs` A)) e. CC /\ ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A))))
50	24, 49	sylbir 218	. . 3 |- (A =/= 0 -> (((*` A) / (abs` A)) e. CC /\ ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A))))
51		fveq2 4681	. . . . . 6 |- (x = ((*` A) / (abs` A)) -> (abs` x) = (abs` ((*` A) / (abs` A))))
52	51	eqeq1d 1892	. . . . 5 |- (x = ((*` A) / (abs` A)) -> ((abs` x) = 1 <-> (abs` ((*` A) / (abs` A))) = 1))
53		opreq1 4889	. . . . . 6 |- (x = ((*` A) / (abs` A)) -> (x x. A) = (((*` A) / (abs` A)) x. A))
54	53	eqeq2d 1895	. . . . 5 |- (x = ((*` A) / (abs` A)) -> ((abs` A) = (x x. A) <-> (abs` A) = (((*` A) / (abs` A)) x. A)))
55	52, 54	anbi12d 690	. . . 4 |- (x = ((*` A) / (abs` A)) -> (((abs` x) = 1 /\ (abs` A) = (x x. A)) <-> ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A))))
56	55	rcla4ev 2381	. . 3 |- ((((*` A) / (abs` A)) e. CC /\ ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A))) -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
57	50, 56	syl 12	. 2 |- (A =/= 0 -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
58	23, 57	pm2.61ine 2089	1 |- E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447   <_ cle 6448  2c2 7145  ^cexp 7811  *ccj 7999  abscabs 8000

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-sup 5664  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  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004

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