gzrngunit - Metamath Proof Explorer

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Theorem gzrngunit 21645

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 21633	. . . . 5 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
2		gzrng.1	. . . . . 6 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
3	2	subrgbas 20742	. . . . 5 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] = (Base‘𝑍))
4	1, 3	ax-mp 5	. . . 4 ⊢ ℤ[i] = (Base‘𝑍)
5		eqid 2762	. . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
6	4, 5	unitcl 20515	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7		eqid 2762	. . . . . . . . . . . 12 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
8		eqid 2762	. . . . . . . . . . . 12 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
9	2, 7, 5, 8	subrginv 20749	. . . . . . . . . . 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 17026	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
12	6, 11	syl 18	. . . . . . . . . . 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 15526	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17		0lt1 11761	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
18	17	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
19	2	gzrngunitlem 21644	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
20	13, 15, 16, 18, 19	ltletrd 11395	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
21	20	gt0ne0d 11803	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
22	12	abs00ad 15377	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
23	22	necon3bid 3001	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
24	21, 23	mpbid 235	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25		cnfldinv 21615	. . . . . . . . . . 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 2799	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((inv_r‘𝑍)‘𝐴) = (1 / 𝐴))
28	2	subrgring 20735	. . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
29	1, 28	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑍 ∈ Ring
30	5, 8	unitinvcl 20530	. . . . . . . . . 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 2863	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
33	2	gzrngunitlem 21644	. . . . . . . 8 ⊢ ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
34	32, 33	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35		1cnd 11227	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
36	35, 12, 24	absdivd 15545	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
37	34, 36	breqtrd 5135	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38		1div1e1 11930	. . . . . 6 ⊢ (1 / 1) = 1
39		abs1 15384	. . . . . . . 8 ⊢ (abs‘1) = 1
40	39	eqcomi 2771	. . . . . . 7 ⊢ 1 = (abs‘1)
41	40	oveq1i 7426	. . . . . 6 ⊢ (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
42	37, 38, 41	3brtr4g 5143	. . . . 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 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 11320	. . . . 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 11194	. . . . . . 7 ⊢ 1 ≠ 0
53	52	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
54	51, 53	eqnetrd 3024	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55		fveq2 6882	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56		abs0 15372	. . . . . . 7 ⊢ (abs‘0) = 0
57	55, 56	eqtrdi 2813	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
58	57	necon3i 2989	. . . . 5 ⊢ ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
59	54, 58	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60		eldifsn 4751	. . . 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 15532	. . . . . . 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 14259	. . . . . . . 8 ⊢ (1↑2) = 1
67	65, 66	eqtrdi 2813	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
68	64, 67	eqtr3d 2799	. . . . . 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 15283	. . . . . 6 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
71	70, 50, 59	divcan3d 12021	. . . . 5 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
72	63, 69, 71	3eqtr2d 2803	. . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (∗‘𝐴))
73		gzcjcl 17030	. . . . 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 2862	. . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ[i])
76		cnfldbas 21588	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
77		cnfld0 21608	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
78		cndrng 21613	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
79	76, 77, 78	drngui 20895	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
80	2, 79, 5, 7	subrgunit 20751	. . . 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 2957 ∖ cdif 3899 {csn 4587 class class class wbr 5107 ‘cfv 6537 (class class class)co 7416 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 · cmul 11130 < clt 11268 ≤ cle 11269 / cdiv 11896 2c2 12320 ↑cexp 14125 ∗ccj 15183 abscabs 15321 ℤ[i]cgz 17023 Basecbs 17303 ↾_s cress 17324 Ringcrg 20371 Unitcui 20495 inv_rcinvr 20527 SubRingcsubrg 20730 ℂ_fldccnfld 21584

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-rp 13043 df-fz 13562 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-gz 17024 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-0g 17528 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-grp 19059 df-minusg 19060 df-subg 19245 df-cmn 19908 df-abl 19909 df-mgp 20273 df-rng 20287 df-ur 20320 df-ring 20373 df-cring 20374 df-oppr 20477 df-dvdsr 20497 df-unit 20498 df-invr 20528 df-dvr 20541 df-subrng 20707 df-subrg 20731 df-drng 20891 df-cnfld 21585

This theorem is used by: zringunit 21678

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