Theorem abs1m 6849

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
abs1m	|- E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A))

Distinct variable group:   x,A

Proof of Theorem abs1m

Step	Hyp	Ref	Expression
1		abs1m.1	. . . . . . . 8 |- A e. CC
2	1	abs00 6785	. . . . . . 7 |- ((abs` A) = 0 <-> A = 0)
3	2	biimpr 152	. . . . . 6 |- (A = 0 -> (abs` A) = 0)
4		opreq2 3960	. . . . . . 7 |- (A = 0 -> (1 x. A) = (1 x. 0))
5		0cn 5308	. . . . . . . 8 |- 0 e. CC
6	5	mulid2 5313	. . . . . . 7 |- (1 x. 0) = 0
7	4, 6	syl6eq 1520	. . . . . 6 |- (A = 0 -> (1 x. A) = 0)
8	3, 7	eqtr4d 1507	. . . . 5 |- (A = 0 -> (abs` A) = (1 x. A))
9		0re 5420	. . . . . . 7 |- 0 e. RR
10		1re 5415	. . . . . . 7 |- 1 e. RR
11		lt01 5661	. . . . . . 7 |- 0 < 1
12	9, 10, 11	ltlei 5562	. . . . . 6 |- 0 <_ 1
13	10	absid 6804	. . . . . 6 |- (0 <_ 1 -> (abs` 1) = 1)
14	12, 13	ax-mp 7	. . . . 5 |- (abs` 1) = 1
15	8, 14	jctil 292	. . . 4 |- (A = 0 -> ((abs` 1) = 1 /\ (abs` A) = (1 x. A)))
16		ax1cn 5249	. . . 4 |- 1 e. CC
17	15, 16	jctil 292	. . 3 |- (A = 0 -> (1 e. CC /\ ((abs` 1) = 1 /\ (abs` A) = (1 x. A))))
18		fveq2 3715	. . . . . 6 |- (x = 1 -> (abs` x) = (abs` 1))
19	18	eqeq1d 1480	. . . . 5 |- (x = 1 -> ((abs` x) = 1 <-> (abs` 1) = 1))
20		opreq1 3959	. . . . . 6 |- (x = 1 -> (x x. A) = (1 x. A))
21	20	eqeq2d 1483	. . . . 5 |- (x = 1 -> ((abs` A) = (x x. A) <-> (abs` A) = (1 x. A)))
22	19, 21	anbi12d 627	. . . 4 |- (x = 1 -> (((abs` x) = 1 /\ (abs` A) = (x x. A)) <-> ((abs` 1) = 1 /\ (abs` A) = (1 x. A))))
23	22	rcla4ev 1873	. . 3 |- ((1 e. CC /\ ((abs` 1) = 1 /\ (abs` A) = (1 x. A))) -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
24	17, 23	syl 10	. 2 |- (A = 0 -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
25	2	necon3bii 1595	. . . 4 |- ((abs` A) =/= 0 <-> A =/= 0)
26	1	cjcl 6707	. . . . . 6 |- (*` A) e. CC
27	1	abscl 6782	. . . . . . 7 |- (abs` A) e. RR
28	27	recn 5294	. . . . . 6 |- (abs` A) e. CC
29	26, 28	divclz 5688	. . . . 5 |- ((abs` A) =/= 0 -> ((*` A) / (abs` A)) e. CC)
30	26, 28	absdivz 6802	. . . . . . 7 |- ((abs` A) =/= 0 -> (abs` ((*` A) / (abs` A))) = ((abs` (*` A)) / (abs` (abs` A))))
31		dividt 5730	. . . . . . . . 9 |- (((abs` A) e. CC /\ (abs` A) =/= 0) -> ((abs` A) / (abs` A)) = 1)
32	28, 31	mpan 694	. . . . . . . 8 |- ((abs` A) =/= 0 -> ((abs` A) / (abs` A)) = 1)
33	1	abscj 6788	. . . . . . . . 9 |- (abs` (*` A)) = (abs` A)
34		absidmt 6838	. . . . . . . . . 10 |- (A e. CC -> (abs` (abs` A)) = (abs` A))
35	1, 34	ax-mp 7	. . . . . . . . 9 |- (abs` (abs` A)) = (abs` A)
36	33, 35	opreq12i 3964	. . . . . . . 8 |- ((abs` (*` A)) / (abs` (abs` A))) = ((abs` A) / (abs` A))
37	32, 36	syl5eq 1516	. . . . . . 7 |- ((abs` A) =/= 0 -> ((abs` (*` A)) / (abs` (abs` A))) = 1)
38	30, 37	eqtrd 1504	. . . . . 6 |- ((abs` A) =/= 0 -> (abs` ((*` A) / (abs` A))) = 1)
39	1	absvalsq 6780	. . . . . . . . 9 |- ((abs` A)^2) = (A x. (*` A))
40	28	sqval 6552	. . . . . . . . 9 |- ((abs` A)^2) = ((abs` A) x. (abs` A))
41	1, 26	mulcom 5303	. . . . . . . . 9 |- (A x. (*` A)) = ((*` A) x. A)
42	39, 40, 41	3eqtr3 1500	. . . . . . . 8 |- ((abs` A) x. (abs` A)) = ((*` A) x. A)
43	26, 1	mulcl 5301	. . . . . . . . 9 |- ((*` A) x. A) e. CC
44	43, 28, 28	divmulz 5683	. . . . . . . 8 |- ((abs` A) =/= 0 -> ((((*` A) x. A) / (abs` A)) = (abs` A) <-> ((abs` A) x. (abs` A)) = ((*` A) x. A)))
45	42, 44	mpbiri 194	. . . . . . 7 |- ((abs` A) =/= 0 -> (((*` A) x. A) / (abs` A)) = (abs` A))
46	26, 1, 28	3pm3.2i 817	. . . . . . . 8 |- ((*` A) e. CC /\ A e. CC /\ (abs` A) e. CC)
47		div23t 5713	. . . . . . . 8 |- ((((*` A) e. CC /\ A e. CC /\ (abs` A) e. CC) /\ (abs` A) =/= 0) -> (((*` A) x. A) / (abs` A)) = (((*` A) / (abs` A)) x. A))
48	46, 47	mpan 694	. . . . . . 7 |- ((abs` A) =/= 0 -> (((*` A) x. A) / (abs` A)) = (((*` A) / (abs` A)) x. A))
49	45, 48	eqtr3d 1506	. . . . . 6 |- ((abs` A) =/= 0 -> (abs` A) = (((*` A) / (abs` A)) x. A))
50	38, 49	jca 288	. . . . 5 |- ((abs` A) =/= 0 -> ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A)))
51	29, 50	jca 288	. . . 4 |- ((abs` A) =/= 0 -> (((*` A) / (abs` A)) e. CC /\ ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A))))
52	25, 51	sylbir 201	. . 3 |- (A =/= 0 -> (((*` A) / (abs` A)) e. CC /\ ((abs` ((*` A) / (abs` A))) = 1 /\ (abs` A) = (((*` A) / (abs` A)) x. A))))
53		fveq2 3715	. . . . . 6 |- (x = ((*` A) / (abs` A)) -> (abs` x) = (abs` ((*` A) / (abs` A))))
54	53	eqeq1d 1480	. . . . 5 |- (x = ((*` A) / (abs` A)) -> ((abs` x) = 1 <-> (abs` ((*` A) / (abs` A))) = 1))
55		opreq1 3959	. . . . . 6 |- (x = ((*` A) / (abs` A)) -> (x x. A) = (((*` A) / (abs` A)) x. A))
56	55	eqeq2d 1483	. . . . 5 |- (x = ((*` A) / (abs` A)) -> ((abs` A) = (x x. A) <-> (abs` A) = (((*` A) / (abs` A)) x. A)))
57	54, 56	anbi12d 627	. . . 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))))
58	57	rcla4ev 1873	. . 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)))
59	52, 58	syl 10	. 2 |- (A =/= 0 -> E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A)))
60	24, 59	pm2.61ine 1631	1 |- E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A))

Syntax hints:   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582  E.wrex 1643   class class class wbr 2614  ` cfv 3177  (class class class)co 3954  CCcc 5212  0cc0 5214  1c1 5215   x. cmul 5219   / cdiv 5274   <_ cle 5275  2c2 5916  ^cexp 6508  *ccj 6688  abscabs 6689

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  <