gzrngunit - Metamath Proof Explorer

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Theorem gzrngunit 21678

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 21666	. . . . 5 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
2		gzrng.1	. . . . . 6 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
3	2	subrgbas 20772	. . . . 5 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] = (Base‘𝑍))
4	1, 3	ax-mp 5	. . . 4 ⊢ ℤ[i] = (Base‘𝑍)
5		eqid 2760	. . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
6	4, 5	unitcl 20544	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7		eqid 2760	. . . . . . . . . . . 12 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
8		eqid 2760	. . . . . . . . . . . 12 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
9	2, 7, 5, 8	subrginv 20779	. . . . . . . . . . 11 ⊢ ((ℤ[i] ∈ (SubRing‘ℂ_fld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((inv_r‘ℂ_fld)‘𝐴) = ((inv_r‘𝑍)‘𝐴))
10	1, 9	mpan 703	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘ℂ_fld)‘𝐴) = ((inv_r‘𝑍)‘𝐴))
11		gzcn 17049	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
12	6, 11	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13		0red 11260	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14		1re 11257	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
16	12	abscld 15551	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17		0lt1 11785	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
18	17	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
19	2	gzrngunitlem 21677	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
20	13, 15, 16, 18, 19	ltletrd 11419	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
21	20	gt0ne0d 11827	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
22	12	abs00ad 15402	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
23	22	necon3bid 2999	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
24	21, 23	mpbid 235	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25		cnfldinv 21648	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
26	12, 24, 25	syl2anc 596	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
27	10, 26	eqtr3d 2797	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) = (1 / 𝐴))
28	2	subrgring 20765	. . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
29	1, 28	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑍 ∈ Ring
30	5, 8	unitinvcl 20559	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((inv_r‘𝑍)‘𝐴) ∈ (Unit‘𝑍))
31	29, 30	mpan 703	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) ∈ (Unit‘𝑍))
32	27, 31	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
33	2	gzrngunitlem 21677	. . . . . . . 8 ⊢ ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
34	32, 33	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35		1cnd 11251	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
36	35, 12, 24	absdivd 15570	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
37	34, 36	breqtrd 5131	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38		1div1e1 11954	. . . . . 6 ⊢ (1 / 1) = 1
39		abs1 15409	. . . . . . . 8 ⊢ (abs‘1) = 1
40	39	eqcomi 2769	. . . . . . 7 ⊢ 1 = (abs‘1)
41	40	oveq1i 7426	. . . . . 6 ⊢ (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
42	37, 38, 41	3brtr4g 5139	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43		lerec 12147	. . . . . 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 852	. . . . 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 11344	. . . . 5 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
47	16, 14, 46	sylancl 598	. . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
48	45, 19, 47	mpbir2and 726	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
49	6, 48	jca 521	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
50	11	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51		simpr 490	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52		ax-1ne0 11218	. . . . . . 7 ⊢ 1 ≠ 0
53	52	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
54	51, 53	eqnetrd 3022	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55		fveq2 6881	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56		abs0 15397	. . . . . . 7 ⊢ (abs‘0) = 0
57	55, 56	eqtrdi 2811	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
58	57	necon3i 2987	. . . . 5 ⊢ ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
59	54, 58	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60		eldifsn 4748	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
61	50, 59, 60	sylanbrc 595	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62		simpl 488	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
63	50, 59, 25	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
64	50	absvalsqd 15557	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
65	51	oveq1d 7431	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66		sq1 14284	. . . . . . . 8 ⊢ (1↑2) = 1
67	65, 66	eqtrdi 2811	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
68	64, 67	eqtr3d 2797	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
69	68	oveq1d 7431	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
70	50	cjcld 15308	. . . . . 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 2801	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (∗‘𝐴))
73		gzcjcl 17053	. . . . 5 ⊢ (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
74	73	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
75	72, 74	eqeltrd 2860	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i])
76		cnfldbas 21621	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
77		cnfld0 21641	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
78		cndrng 21646	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
79	76, 77, 78	drngui 20925	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
80	2, 79, 5, 7	subrgunit 20781	. . . 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 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∖ cdif 3896 {csn 4584 class class class wbr 5103 ‘cfv 6535 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 · cmul 11154 < clt 11292 ≤ cle 11293 / cdiv 11920 2c2 12344 ↑cexp 14150 ∗ccj 15208 abscabs 15346 ℤ[i]cgz 17046 Basecbs 17326 ↾_s cress 17347 Ringcrg 20398 Unitcui 20524 inv_rcinvr 20556 SubRingcsubrg 20760 ℂ_fldccnfld 21617

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8229 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9419 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-rp 13068 df-fz 13587 df-seq 14091 df-exp 14151 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-gz 17047 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-0g 17551 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-grp 19086 df-minusg 19087 df-subg 19272 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-cring 20401 df-oppr 20506 df-dvdsr 20526 df-unit 20527 df-invr 20557 df-dvr 20570 df-subrng 20737 df-subrg 20761 df-drng 20921 df-cnfld 21618

This theorem is used by: zringunit 21711

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