zringunit - Metamath Proof Explorer

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Theorem zringunit 21028

Description: The units of ℤ are the integers with norm 1, i.e. 1 and -1. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)

**Assertion**
Ref	Expression
zringunit	⊢ (𝐴 ∈ (Unit‘ℤ_ring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))

**Proof of Theorem zringunit**
Step	Hyp	Ref	Expression
1		zringbas 21016	. . . 4 ⊢ ℤ = (Base‘ℤ_ring)
2		eqid 2733	. . . 4 ⊢ (Unit‘ℤ_ring) = (Unit‘ℤ_ring)
3	1, 2	unitcl 20182	. . 3 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) → 𝐴 ∈ ℤ)
4		zsubrg 20991	. . . . . . 7 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
5		zgz 16863	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℤ[i])
6	5	ssriv 3986	. . . . . . 7 ⊢ ℤ ⊆ ℤ[i]
7		gzsubrg 20992	. . . . . . . 8 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
8		eqid 2733	. . . . . . . . 9 ⊢ (ℂ_fld ↾_s ℤ[i]) = (ℂ_fld ↾_s ℤ[i])
9	8	subsubrg 20383	. . . . . . . 8 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → (ℤ ∈ (SubRing‘(ℂ_fld ↾_s ℤ[i])) ↔ (ℤ ∈ (SubRing‘ℂ_fld) ∧ ℤ ⊆ ℤ[i])))
10	7, 9	ax-mp 5	. . . . . . 7 ⊢ (ℤ ∈ (SubRing‘(ℂ_fld ↾_s ℤ[i])) ↔ (ℤ ∈ (SubRing‘ℂ_fld) ∧ ℤ ⊆ ℤ[i]))
11	4, 6, 10	mpbir2an 710	. . . . . 6 ⊢ ℤ ∈ (SubRing‘(ℂ_fld ↾_s ℤ[i]))
12		df-zring 21011	. . . . . . . 8 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
13		ressabs 17191	. . . . . . . . 9 ⊢ ((ℤ[i] ∈ (SubRing‘ℂ_fld) ∧ ℤ ⊆ ℤ[i]) → ((ℂ_fld ↾_s ℤ[i]) ↾_s ℤ) = (ℂ_fld ↾_s ℤ))
14	7, 6, 13	mp2an 691	. . . . . . . 8 ⊢ ((ℂ_fld ↾_s ℤ[i]) ↾_s ℤ) = (ℂ_fld ↾_s ℤ)
15	12, 14	eqtr4i 2764	. . . . . . 7 ⊢ ℤ_ring = ((ℂ_fld ↾_s ℤ[i]) ↾_s ℤ)
16		eqid 2733	. . . . . . 7 ⊢ (Unit‘(ℂ_fld ↾_s ℤ[i])) = (Unit‘(ℂ_fld ↾_s ℤ[i]))
17	15, 16, 2	subrguss 20371	. . . . . 6 ⊢ (ℤ ∈ (SubRing‘(ℂ_fld ↾_s ℤ[i])) → (Unit‘ℤ_ring) ⊆ (Unit‘(ℂ_fld ↾_s ℤ[i])))
18	11, 17	ax-mp 5	. . . . 5 ⊢ (Unit‘ℤ_ring) ⊆ (Unit‘(ℂ_fld ↾_s ℤ[i]))
19	18	sseli 3978	. . . 4 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) → 𝐴 ∈ (Unit‘(ℂ_fld ↾_s ℤ[i])))
20	8	gzrngunit 21004	. . . . 5 ⊢ (𝐴 ∈ (Unit‘(ℂ_fld ↾_s ℤ[i])) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
21	20	simprbi 498	. . . 4 ⊢ (𝐴 ∈ (Unit‘(ℂ_fld ↾_s ℤ[i])) → (abs‘𝐴) = 1)
22	19, 21	syl 17	. . 3 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) → (abs‘𝐴) = 1)
23	3, 22	jca 513	. 2 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) → (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
24		zcn 12560	. . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
25	24	adantr 482	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
26		simpr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
27		ax-1ne0 11176	. . . . . . 7 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
29	26, 28	eqnetrd 3009	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
30		fveq2 6889	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
31		abs0 15229	. . . . . . 7 ⊢ (abs‘0) = 0
32	30, 31	eqtrdi 2789	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
33	32	necon3i 2974	. . . . 5 ⊢ ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
34	29, 33	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
35		eldifsn 4790	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
36	25, 34, 35	sylanbrc 584	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
37		simpl 484	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ)
38		cnfldinv 20969	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
39	25, 34, 38	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
40		zre 12559	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
41	40	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℝ)
42		absresq 15246	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2))
43	41, 42	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴↑2))
44	26	oveq1d 7421	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
45		sq1 14156	. . . . . . . 8 ⊢ (1↑2) = 1
46	44, 45	eqtrdi 2789	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
47	25	sqvald 14105	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴↑2) = (𝐴 · 𝐴))
48	43, 46, 47	3eqtr3rd 2782	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴 · 𝐴) = 1)
49		1cnd 11206	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℂ)
50	49, 25, 25, 34	divmuld 12009	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((1 / 𝐴) = 𝐴 ↔ (𝐴 · 𝐴) = 1))
51	48, 50	mpbird 257	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = 𝐴)
52	39, 51	eqtrd 2773	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = 𝐴)
53	52, 37	eqeltrd 2834	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ)
54		cnfldbas 20941	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
55		cnfld0 20962	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
56		cndrng 20967	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
57	54, 55, 56	drngui 20314	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
58		eqid 2733	. . . . 5 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
59	12, 57, 2, 58	subrgunit 20374	. . . 4 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → (𝐴 ∈ (Unit‘ℤ_ring) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ ∧ ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ)))
60	4, 59	ax-mp 5	. . 3 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ ∧ ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ))
61	36, 37, 53, 60	syl3anbrc 1344	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘ℤ_ring))
62	23, 61	impbii 208	1 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 · cmul 11112 / cdiv 11868 2c2 12264 ℤcz 12555 ↑cexp 14024 abscabs 15178 ℤ[i]cgz 16859 ↾_s cress 17170 Unitcui 20162 inv_rcinvr 20194 SubRingcsubrg 20352 ℂ_fldccnfld 20937 ℤ_ringczring 21010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-fz 13482 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-gz 16860 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-minusg 18820 df-subg 18998 df-cmn 19645 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-dvr 20208 df-drng 20310 df-subrg 20354 df-cnfld 20938 df-zring 21011

This theorem is referenced by: zringndrg 21030 prmirredlem 21034 qqhval2lem 32950

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