zringunit - Metamath Proof Explorer

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Theorem zringunit 21042

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 21029	. . . 4 ⊢ ℤ = (Base‘ℤ_ring)
2		eqid 2732	. . . 4 ⊢ (Unit‘ℤ_ring) = (Unit‘ℤ_ring)
3	1, 2	unitcl 20193	. . 3 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) → 𝐴 ∈ ℤ)
4		zsubrg 21004	. . . . . . 7 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
5		zgz 16868	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℤ[i])
6	5	ssriv 3986	. . . . . . 7 ⊢ ℤ ⊆ ℤ[i]
7		gzsubrg 21005	. . . . . . . 8 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
8		eqid 2732	. . . . . . . . 9 ⊢ (ℂ_fld ↾_s ℤ[i]) = (ℂ_fld ↾_s ℤ[i])
9	8	subsubrg 20349	. . . . . . . 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 709	. . . . . 6 ⊢ ℤ ∈ (SubRing‘(ℂ_fld ↾_s ℤ[i]))
12		df-zring 21024	. . . . . . . 8 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
13		ressabs 17196	. . . . . . . . 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 690	. . . . . . . 8 ⊢ ((ℂ_fld ↾_s ℤ[i]) ↾_s ℤ) = (ℂ_fld ↾_s ℤ)
15	12, 14	eqtr4i 2763	. . . . . . 7 ⊢ ℤ_ring = ((ℂ_fld ↾_s ℤ[i]) ↾_s ℤ)
16		eqid 2732	. . . . . . 7 ⊢ (Unit‘(ℂ_fld ↾_s ℤ[i])) = (Unit‘(ℂ_fld ↾_s ℤ[i]))
17	15, 16, 2	subrguss 20338	. . . . . 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 21017	. . . . 5 ⊢ (𝐴 ∈ (Unit‘(ℂ_fld ↾_s ℤ[i])) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
21	20	simprbi 497	. . . 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 512	. 2 ⊢ (𝐴 ∈ (Unit‘ℤ_ring) → (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
24		zcn 12565	. . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
25	24	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
26		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
27		ax-1ne0 11181	. . . . . . 7 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
29	26, 28	eqnetrd 3008	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
30		fveq2 6891	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
31		abs0 15234	. . . . . . 7 ⊢ (abs‘0) = 0
32	30, 31	eqtrdi 2788	. . . . . 6 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
33	32	necon3i 2973	. . . . 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 583	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
37		simpl 483	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ)
38		cnfldinv 20982	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
39	25, 34, 38	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = (1 / 𝐴))
40		zre 12564	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
41	40	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℝ)
42		absresq 15251	. . . . . . . 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 7426	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
45		sq1 14161	. . . . . . . 8 ⊢ (1↑2) = 1
46	44, 45	eqtrdi 2788	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
47	25	sqvald 14110	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴↑2) = (𝐴 · 𝐴))
48	43, 46, 47	3eqtr3rd 2781	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴 · 𝐴) = 1)
49		1cnd 11211	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℂ)
50	49, 25, 25, 34	divmuld 12014	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((1 / 𝐴) = 𝐴 ↔ (𝐴 · 𝐴) = 1))
51	48, 50	mpbird 256	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = 𝐴)
52	39, 51	eqtrd 2772	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) = 𝐴)
53	52, 37	eqeltrd 2833	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((inv_r‘ℂ_fld)‘𝐴) ∈ ℤ)
54		cnfldbas 20954	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
55		cnfld0 20975	. . . . . 6 ⊢ 0 = (0_g‘ℂ_fld)
56		cndrng 20980	. . . . . 6 ⊢ ℂ_fld ∈ DivRing
57	54, 55, 56	drngui 20367	. . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
58		eqid 2732	. . . . 5 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
59	12, 57, 2, 58	subrgunit 20341	. . . 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 1343	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 / cdiv 11873 2c2 12269 ℤcz 12560 ↑cexp 14029 abscabs 15183 ℤ[i]cgz 16864 ↾_s cress 17175 Unitcui 20173 inv_rcinvr 20205 SubRingcsubrg 20319 ℂ_fldccnfld 20950 ℤ_ringczring 21023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-fz 13487 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-gz 16865 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-subg 19005 df-cmn 19652 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-subrg 20321 df-drng 20363 df-cnfld 20951 df-zring 21024

This theorem is referenced by: zringndrg 21044 prmirredlem 21048 qqhval2lem 33030

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