gzrngunit - Metamath Proof Explorer

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Theorem gzrngunit 21399

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 21387	. . . . 5 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
2		gzrng.1	. . . . . 6 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
3	2	subrgbas 20539	. . . . 5 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] = (Base‘𝑍))
4	1, 3	ax-mp 5	. . . 4 ⊢ ℤ[i] = (Base‘𝑍)
5		eqid 2735	. . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
6	4, 5	unitcl 20333	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7		eqid 2735	. . . . . . . . . . . 12 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
8		eqid 2735	. . . . . . . . . . . 12 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
9	2, 7, 5, 8	subrginv 20546	. . . . . . . . . . 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 16950	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
12	6, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13		0red 11236	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14		1re 11233	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
16	12	abscld 15453	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17		0lt1 11757	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
18	17	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
19	2	gzrngunitlem 21398	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
20	13, 15, 16, 18, 19	ltletrd 11393	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
21	20	gt0ne0d 11799	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
22	12	abs00ad 15307	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
23	22	necon3bid 2976	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
24	21, 23	mpbid 232	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25		cnfldinv 21363	. . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) = (1 / 𝐴))
28	2	subrgring 20532	. . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
29	1, 28	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑍 ∈ Ring
30	5, 8	unitinvcl 20348	. . . . . . . . . 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 2835	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
33	2	gzrngunitlem 21398	. . . . . . . 8 ⊢ ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
34	32, 33	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35		1cnd 11228	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
36	35, 12, 24	absdivd 15472	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
37	34, 36	breqtrd 5145	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38		1div1e1 11930	. . . . . 6 ⊢ (1 / 1) = 1
39		abs1 15314	. . . . . . . 8 ⊢ (abs‘1) = 1
40	39	eqcomi 2744	. . . . . . 7 ⊢ 1 = (abs‘1)
41	40	oveq1i 7413	. . . . . 6 ⊢ (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
42	37, 38, 41	3brtr4g 5153	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43		lerec 12123	. . . . . 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 838	. . . . 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 11318	. . . . 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 11196	. . . . . . 7 ⊢ 1 ≠ 0
53	52	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
54	51, 53	eqnetrd 2999	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55		fveq2 6875	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56		abs0 15302	. . . . . . 7 ⊢ (abs‘0) = 0
57	55, 56	eqtrdi 2786	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
58	57	necon3i 2964	. . . . 5 ⊢ ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
59	54, 58	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60		eldifsn 4762	. . . 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 15459	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
65	51	oveq1d 7418	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66		sq1 14211	. . . . . . . 8 ⊢ (1↑2) = 1
67	65, 66	eqtrdi 2786	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
68	64, 67	eqtr3d 2772	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
69	68	oveq1d 7418	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
70	50	cjcld 15213	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
71	70, 50, 59	divcan3d 12020	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
72	63, 69, 71	3eqtr2d 2776	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (∗‘𝐴))
73		gzcjcl 16954	. . . . 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 2834	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i])
76		cnfldbas 21317	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
77		cnfld0 21353	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
78		cndrng 21359	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
79	76, 77, 78	drngui 20693	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
80	2, 79, 5, 7	subrgunit 20548	. . . 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 1344	. 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 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 {csn 4601 class class class wbr 5119 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 · cmul 11132 < clt 11267 ≤ cle 11268 / cdiv 11892 2c2 12293 ↑cexp 14077 ∗ccj 15113 abscabs 15251 ℤ[i]cgz 16947 Basecbs 17226 ↾_s cress 17249 Ringcrg 20191 Unitcui 20313 inv_rcinvr 20345 SubRingcsubrg 20527 ℂ_fldccnfld 21313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-rp 13007 df-fz 13523 df-seq 14018 df-exp 14078 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-gz 16948 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-0g 17453 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-grp 18917 df-minusg 18918 df-subg 19104 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-invr 20346 df-dvr 20359 df-subrng 20504 df-subrg 20528 df-drng 20689 df-cnfld 21314

This theorem is referenced by: zringunit 21425

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