gzrngunit - Metamath Proof Explorer

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Theorem gzrngunit 21592

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 21580	. . . . 5 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
2		gzrng.1	. . . . . 6 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
3	2	subrgbas 20689	. . . . 5 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] = (Base‘𝑍))
4	1, 3	ax-mp 5	. . . 4 ⊢ ℤ[i] = (Base‘𝑍)
5		eqid 2763	. . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
6	4, 5	unitcl 20462	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7		eqid 2763	. . . . . . . . . . . 12 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
8		eqid 2763	. . . . . . . . . . . 12 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
9	2, 7, 5, 8	subrginv 20696	. . . . . . . . . . 11 ⊢ ((ℤ[i] ∈ (SubRing‘ℂ_fld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((inv_r‘ℂ_fld)‘𝐴) = ((inv_r‘𝑍)‘𝐴))
10	1, 9	mpan 702	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘ℂ_fld)‘𝐴) = ((inv_r‘𝑍)‘𝐴))
11		gzcn 16996	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
12	6, 11	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13		0red 11215	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14		1re 11212	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
16	12	abscld 15495	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17		0lt1 11740	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
18	17	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
19	2	gzrngunitlem 21591	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
20	13, 15, 16, 18, 19	ltletrd 11374	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
21	20	gt0ne0d 11782	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
22	12	abs00ad 15346	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
23	22	necon3bid 3002	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
24	21, 23	mpbid 235	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25		cnfldinv 21562	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
26	12, 24, 25	syl2anc 595	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
27	10, 26	eqtr3d 2800	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) = (1 / 𝐴))
28	2	subrgring 20682	. . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
29	1, 28	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑍 ∈ Ring
30	5, 8	unitinvcl 20477	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((inv_r‘𝑍)‘𝐴) ∈ (Unit‘𝑍))
31	29, 30	mpan 702	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) ∈ (Unit‘𝑍))
32	27, 31	eqeltrrd 2864	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
33	2	gzrngunitlem 21591	. . . . . . . 8 ⊢ ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
34	32, 33	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35		1cnd 11206	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
36	35, 12, 24	absdivd 15514	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
37	34, 36	breqtrd 5137	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38		1div1e1 11909	. . . . . 6 ⊢ (1 / 1) = 1
39		abs1 15353	. . . . . . . 8 ⊢ (abs‘1) = 1
40	39	eqcomi 2772	. . . . . . 7 ⊢ 1 = (abs‘1)
41	40	oveq1i 7420	. . . . . 6 ⊢ (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
42	37, 38, 41	3brtr4g 5145	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43		lerec 12102	. . . . . 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 851	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
45	42, 44	mpbird 260	. . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≤ 1)
46		letri3 11299	. . . . 5 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
47	16, 14, 46	sylancl 597	. . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
48	45, 19, 47	mpbir2and 725	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
49	6, 48	jca 520	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
50	11	adantr 485	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51		simpr 489	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52		ax-1ne0 11173	. . . . . . 7 ⊢ 1 ≠ 0
53	52	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
54	51, 53	eqnetrd 3025	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55		fveq2 6881	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56		abs0 15341	. . . . . . 7 ⊢ (abs‘0) = 0
57	55, 56	eqtrdi 2814	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
58	57	necon3i 2990	. . . . 5 ⊢ ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
59	54, 58	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60		eldifsn 4753	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
61	50, 59, 60	sylanbrc 594	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62		simpl 487	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
63	50, 59, 25	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
64	50	absvalsqd 15501	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
65	51	oveq1d 7425	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66		sq1 14236	. . . . . . . 8 ⊢ (1↑2) = 1
67	65, 66	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
68	64, 67	eqtr3d 2800	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
69	68	oveq1d 7425	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
70	50	cjcld 15252	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
71	70, 50, 59	divcan3d 12000	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
72	63, 69, 71	3eqtr2d 2804	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (∗‘𝐴))
73		gzcjcl 17000	. . . . 5 ⊢ (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
74	73	adantr 485	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
75	72, 74	eqeltrd 2863	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i])
76		cnfldbas 21535	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
77		cnfld0 21555	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
78		cndrng 21560	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
79	76, 77, 78	drngui 20842	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
80	2, 79, 5, 7	subrgunit 20698	. . . 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 1362	. 2 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘𝑍))
83	49, 82	impbii 212	1 ⊢ (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3902 {csn 4589 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℂcc 11102 ℝcr 11103 0cc0 11104 1c1 11105 · cmul 11109 < clt 11247 ≤ cle 11248 / cdiv 11875 2c2 12299 ↑cexp 14102 ∗ccj 15152 abscabs 15290 ℤ[i]cgz 16993 Basecbs 17273 ↾_s cress 17294 Ringcrg 20319 Unitcui 20442 inv_rcinvr 20474 SubRingcsubrg 20677 ℂ_fldccnfld 21531

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-rp 13021 df-fz 13540 df-seq 14043 df-exp 14103 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-gz 16994 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-0g 17498 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-grp 19007 df-minusg 19008 df-subg 19193 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-cring 20322 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-dvr 20488 df-subrng 20654 df-subrg 20678 df-drng 20838 df-cnfld 21532

This theorem is used by: zringunit 21625

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