gzrngunit - Metamath Proof Explorer

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Theorem gzrngunit 21468

Description: The units on ℤ[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.)

**Hypothesis**
Ref	Expression
gzrng.1	⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])

**Assertion**
Ref	Expression
gzrngunit	⊢ (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))

**Proof of Theorem gzrngunit**
Step	Hyp	Ref	Expression
1		gzsubrg 21456	. . . . 5 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
2		gzrng.1	. . . . . 6 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
3	2	subrgbas 20597	. . . . 5 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] = (Base‘𝑍))
4	1, 3	ax-mp 5	. . . 4 ⊢ ℤ[i] = (Base‘𝑍)
5		eqid 2734	. . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
6	4, 5	unitcl 20391	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7		eqid 2734	. . . . . . . . . . . 12 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
8		eqid 2734	. . . . . . . . . . . 12 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
9	2, 7, 5, 8	subrginv 20604	. . . . . . . . . . 11 ⊢ ((ℤ[i] ∈ (SubRing‘ℂ_fld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((inv_r‘ℂ_fld)‘𝐴) = ((inv_r‘𝑍)‘𝐴))
10	1, 9	mpan 690	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘ℂ_fld)‘𝐴) = ((inv_r‘𝑍)‘𝐴))
11		gzcn 16965	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
12	6, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13		0red 11261	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14		1re 11258	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
16	12	abscld 15471	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17		0lt1 11782	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
18	17	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
19	2	gzrngunitlem 21467	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
20	13, 15, 16, 18, 19	ltletrd 11418	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
21	20	gt0ne0d 11824	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
22	12	abs00ad 15325	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
23	22	necon3bid 2982	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
24	21, 23	mpbid 232	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25		cnfldinv 21432	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
26	12, 24, 25	syl2anc 584	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
27	10, 26	eqtr3d 2776	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) = (1 / 𝐴))
28	2	subrgring 20590	. . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
29	1, 28	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑍 ∈ Ring
30	5, 8	unitinvcl 20406	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((inv_r‘𝑍)‘𝐴) ∈ (Unit‘𝑍))
31	29, 30	mpan 690	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) ∈ (Unit‘𝑍))
32	27, 31	eqeltrrd 2839	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
33	2	gzrngunitlem 21467	. . . . . . . 8 ⊢ ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
34	32, 33	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35		1cnd 11253	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
36	35, 12, 24	absdivd 15490	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
37	34, 36	breqtrd 5173	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38		1div1e1 11955	. . . . . 6 ⊢ (1 / 1) = 1
39		abs1 15332	. . . . . . . 8 ⊢ (abs‘1) = 1
40	39	eqcomi 2743	. . . . . . 7 ⊢ 1 = (abs‘1)
41	40	oveq1i 7440	. . . . . 6 ⊢ (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
42	37, 38, 41	3brtr4g 5181	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43		lerec 12148	. . . . . 6 ⊢ ((((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴)) ∧ (1 ∈ ℝ ∧ 0 < 1)) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
44	16, 20, 15, 18, 43	syl22anc 839	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
45	42, 44	mpbird 257	. . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≤ 1)
46		letri3 11343	. . . . 5 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
47	16, 14, 46	sylancl 586	. . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
48	45, 19, 47	mpbir2and 713	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
49	6, 48	jca 511	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
50	11	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52		ax-1ne0 11221	. . . . . . 7 ⊢ 1 ≠ 0
53	52	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
54	51, 53	eqnetrd 3005	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55		fveq2 6906	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56		abs0 15320	. . . . . . 7 ⊢ (abs‘0) = 0
57	55, 56	eqtrdi 2790	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
58	57	necon3i 2970	. . . . 5 ⊢ ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
59	54, 58	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60		eldifsn 4790	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
61	50, 59, 60	sylanbrc 583	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62		simpl 482	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
63	50, 59, 25	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
64	50	absvalsqd 15477	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
65	51	oveq1d 7445	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66		sq1 14230	. . . . . . . 8 ⊢ (1↑2) = 1
67	65, 66	eqtrdi 2790	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
68	64, 67	eqtr3d 2776	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
69	68	oveq1d 7445	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
70	50	cjcld 15231	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
71	70, 50, 59	divcan3d 12045	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
72	63, 69, 71	3eqtr2d 2780	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (∗‘𝐴))
73		gzcjcl 16969	. . . . 5 ⊢ (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
74	73	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
75	72, 74	eqeltrd 2838	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i])
76		cnfldbas 21385	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
77		cnfld0 21422	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
78		cndrng 21428	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
79	76, 77, 78	drngui 20751	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
80	2, 79, 5, 7	subrgunit 20606	. . . 4 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ[i] ∧ ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i])))
81	1, 80	ax-mp 5	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ[i] ∧ ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i]))
82	61, 62, 75, 81	syl3anbrc 1342	. 2 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘𝑍))
83	49, 82	impbii 209	1 ⊢ (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 < clt 11292 ≤ cle 11293 / cdiv 11917 2c2 12318 ↑cexp 14098 ∗ccj 15131 abscabs 15269 ℤ[i]cgz 16962 Basecbs 17244 ↾_s cress 17273 Ringcrg 20250 Unitcui 20371 inv_rcinvr 20403 SubRingcsubrg 20585 ℂ_fldccnfld 21381

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-gz 16963 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-subrng 20562 df-subrg 20586 df-drng 20747 df-cnfld 21382

This theorem is referenced by: zringunit 21494

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