Theorem abs1m 14075

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. (Contributed by NM, 26-Mar-2005.)

Assertion

Ref	Expression
abs1m	⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem abs1m

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
2		abs0 14025	. . . . . 6 ⊢ (abs‘0) = 0
3	1, 2	syl6eq 2672	. . . . 5 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
4		oveq2 6658	. . . . 5 ⊢ (𝐴 = 0 → (𝑥 · 𝐴) = (𝑥 · 0))
5	3, 4	eqeq12d 2637	. . . 4 ⊢ (𝐴 = 0 → ((abs‘𝐴) = (𝑥 · 𝐴) ↔ 0 = (𝑥 · 0)))
6	5	anbi2d 740	. . 3 ⊢ (𝐴 = 0 → (((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)) ↔ ((abs‘𝑥) = 1 ∧ 0 = (𝑥 · 0))))
7	6	rexbidv 3052	. 2 ⊢ (𝐴 = 0 → (∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)) ↔ ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ 0 = (𝑥 · 0))))
8		simpl 473	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
9	8	cjcld 13936	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
10		abscl 14018	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
11	10	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
12	11	recnd 10068	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ)
13		abs00 14029	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
14	13	necon3bid 2838	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
15	14	biimpar 502	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
16	9, 12, 15	divcld 10801	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((∗‘𝐴) / (abs‘𝐴)) ∈ ℂ)
17		absdiv 14035	. . . . 5 ⊢ (((∗‘𝐴) ∈ ℂ ∧ (abs‘𝐴) ∈ ℂ ∧ (abs‘𝐴) ≠ 0) → (abs‘((∗‘𝐴) / (abs‘𝐴))) = ((abs‘(∗‘𝐴)) / (abs‘(abs‘𝐴))))
18	9, 12, 15, 17	syl3anc 1326	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘((∗‘𝐴) / (abs‘𝐴))) = ((abs‘(∗‘𝐴)) / (abs‘(abs‘𝐴))))
19		abscj 14019	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (abs‘𝐴))
20	19	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(∗‘𝐴)) = (abs‘𝐴))
21		absidm 14063	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘(abs‘𝐴)) = (abs‘𝐴))
22	21	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(abs‘𝐴)) = (abs‘𝐴))
23	20, 22	oveq12d 6668	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘(∗‘𝐴)) / (abs‘(abs‘𝐴))) = ((abs‘𝐴) / (abs‘𝐴)))
24	12, 15	dividd 10799	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴) / (abs‘𝐴)) = 1)
25	18, 23, 24	3eqtrd 2660	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘((∗‘𝐴) / (abs‘𝐴))) = 1)
26	8, 9, 12, 15	divassd 10836	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / (abs‘𝐴)) = (𝐴 · ((∗‘𝐴) / (abs‘𝐴))))
27	12	sqvald 13005	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑2) = ((abs‘𝐴) · (abs‘𝐴)))
28		absvalsq 14020	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
29	28	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
30	27, 29	eqtr3d 2658	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴) · (abs‘𝐴)) = (𝐴 · (∗‘𝐴)))
31	12, 12, 15, 30	mvllmuld 10857	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) = ((𝐴 · (∗‘𝐴)) / (abs‘𝐴)))
32	16, 8	mulcomd 10061	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((∗‘𝐴) / (abs‘𝐴)) · 𝐴) = (𝐴 · ((∗‘𝐴) / (abs‘𝐴))))
33	26, 31, 32	3eqtr4d 2666	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) = (((∗‘𝐴) / (abs‘𝐴)) · 𝐴))
34		fveq2 6191	. . . . . 6 ⊢ (𝑥 = ((∗‘𝐴) / (abs‘𝐴)) → (abs‘𝑥) = (abs‘((∗‘𝐴) / (abs‘𝐴))))
35	34	eqeq1d 2624	. . . . 5 ⊢ (𝑥 = ((∗‘𝐴) / (abs‘𝐴)) → ((abs‘𝑥) = 1 ↔ (abs‘((∗‘𝐴) / (abs‘𝐴))) = 1))
36		oveq1 6657	. . . . . 6 ⊢ (𝑥 = ((∗‘𝐴) / (abs‘𝐴)) → (𝑥 · 𝐴) = (((∗‘𝐴) / (abs‘𝐴)) · 𝐴))
37	36	eqeq2d 2632	. . . . 5 ⊢ (𝑥 = ((∗‘𝐴) / (abs‘𝐴)) → ((abs‘𝐴) = (𝑥 · 𝐴) ↔ (abs‘𝐴) = (((∗‘𝐴) / (abs‘𝐴)) · 𝐴)))
38	35, 37	anbi12d 747	. . . 4 ⊢ (𝑥 = ((∗‘𝐴) / (abs‘𝐴)) → (((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)) ↔ ((abs‘((∗‘𝐴) / (abs‘𝐴))) = 1 ∧ (abs‘𝐴) = (((∗‘𝐴) / (abs‘𝐴)) · 𝐴))))
39	38	rspcev 3309	. . 3 ⊢ ((((∗‘𝐴) / (abs‘𝐴)) ∈ ℂ ∧ ((abs‘((∗‘𝐴) / (abs‘𝐴))) = 1 ∧ (abs‘𝐴) = (((∗‘𝐴) / (abs‘𝐴)) · 𝐴))) → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)))
40	16, 25, 33, 39	syl12anc 1324	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)))
41		ax-icn 9995	. . . 4 ⊢ i ∈ ℂ
42		absi 14026	. . . . 5 ⊢ (abs‘i) = 1
43		it0e0 11254	. . . . . 6 ⊢ (i · 0) = 0
44	43	eqcomi 2631	. . . . 5 ⊢ 0 = (i · 0)
45	42, 44	pm3.2i 471	. . . 4 ⊢ ((abs‘i) = 1 ∧ 0 = (i · 0))
46		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = i → (abs‘𝑥) = (abs‘i))
47	46	eqeq1d 2624	. . . . . 6 ⊢ (𝑥 = i → ((abs‘𝑥) = 1 ↔ (abs‘i) = 1))
48		oveq1 6657	. . . . . . 7 ⊢ (𝑥 = i → (𝑥 · 0) = (i · 0))
49	48	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = i → (0 = (𝑥 · 0) ↔ 0 = (i · 0)))
50	47, 49	anbi12d 747	. . . . 5 ⊢ (𝑥 = i → (((abs‘𝑥) = 1 ∧ 0 = (𝑥 · 0)) ↔ ((abs‘i) = 1 ∧ 0 = (i · 0))))
51	50	rspcev 3309	. . . 4 ⊢ ((i ∈ ℂ ∧ ((abs‘i) = 1 ∧ 0 = (i · 0))) → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ 0 = (𝑥 · 0)))
52	41, 45, 51	mp2an 708	. . 3 ⊢ ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ 0 = (𝑥 · 0))
53	52	a1i 11	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ 0 = (𝑥 · 0)))
54	7, 40, 53	pm2.61ne 2879	1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937  ici 9938   · cmul 9941   / cdiv 10684  2c2 11070  ↑cexp 12860  ∗ccj 13836  abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by: (None)