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Theorem zringunit 19750
Description: The units of 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‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))

Proof of Theorem zringunit
StepHypRef Expression
1 zringbas 19738 . . . 4 ℤ = (Base‘ℤring)
2 eqid 2626 . . . 4 (Unit‘ℤring) = (Unit‘ℤring)
31, 2unitcl 18575 . . 3 (𝐴 ∈ (Unit‘ℤring) → 𝐴 ∈ ℤ)
4 zsubrg 19713 . . . . . . 7 ℤ ∈ (SubRing‘ℂfld)
5 zgz 15556 . . . . . . . 8 (𝑥 ∈ ℤ → 𝑥 ∈ ℤ[i])
65ssriv 3592 . . . . . . 7 ℤ ⊆ ℤ[i]
7 gzsubrg 19714 . . . . . . . 8 ℤ[i] ∈ (SubRing‘ℂfld)
8 eqid 2626 . . . . . . . . 9 (ℂflds ℤ[i]) = (ℂflds ℤ[i])
98subsubrg 18722 . . . . . . . 8 (ℤ[i] ∈ (SubRing‘ℂfld) → (ℤ ∈ (SubRing‘(ℂflds ℤ[i])) ↔ (ℤ ∈ (SubRing‘ℂfld) ∧ ℤ ⊆ ℤ[i])))
107, 9ax-mp 5 . . . . . . 7 (ℤ ∈ (SubRing‘(ℂflds ℤ[i])) ↔ (ℤ ∈ (SubRing‘ℂfld) ∧ ℤ ⊆ ℤ[i]))
114, 6, 10mpbir2an 954 . . . . . 6 ℤ ∈ (SubRing‘(ℂflds ℤ[i]))
12 df-zring 19733 . . . . . . . 8 ring = (ℂflds ℤ)
13 ressabs 15855 . . . . . . . . 9 ((ℤ[i] ∈ (SubRing‘ℂfld) ∧ ℤ ⊆ ℤ[i]) → ((ℂflds ℤ[i]) ↾s ℤ) = (ℂflds ℤ))
147, 6, 13mp2an 707 . . . . . . . 8 ((ℂflds ℤ[i]) ↾s ℤ) = (ℂflds ℤ)
1512, 14eqtr4i 2651 . . . . . . 7 ring = ((ℂflds ℤ[i]) ↾s ℤ)
16 eqid 2626 . . . . . . 7 (Unit‘(ℂflds ℤ[i])) = (Unit‘(ℂflds ℤ[i]))
1715, 16, 2subrguss 18711 . . . . . 6 (ℤ ∈ (SubRing‘(ℂflds ℤ[i])) → (Unit‘ℤring) ⊆ (Unit‘(ℂflds ℤ[i])))
1811, 17ax-mp 5 . . . . 5 (Unit‘ℤring) ⊆ (Unit‘(ℂflds ℤ[i]))
1918sseli 3584 . . . 4 (𝐴 ∈ (Unit‘ℤring) → 𝐴 ∈ (Unit‘(ℂflds ℤ[i])))
208gzrngunit 19726 . . . . 5 (𝐴 ∈ (Unit‘(ℂflds ℤ[i])) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
2120simprbi 480 . . . 4 (𝐴 ∈ (Unit‘(ℂflds ℤ[i])) → (abs‘𝐴) = 1)
2219, 21syl 17 . . 3 (𝐴 ∈ (Unit‘ℤring) → (abs‘𝐴) = 1)
233, 22jca 554 . 2 (𝐴 ∈ (Unit‘ℤring) → (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
24 zcn 11327 . . . . 5 (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
2524adantr 481 . . . 4 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
26 simpr 477 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
27 ax-1ne0 9950 . . . . . . 7 1 ≠ 0
2827a1i 11 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
2926, 28eqnetrd 2863 . . . . 5 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
30 fveq2 6150 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
31 abs0 13954 . . . . . . 7 (abs‘0) = 0
3230, 31syl6eq 2676 . . . . . 6 (𝐴 = 0 → (abs‘𝐴) = 0)
3332necon3i 2828 . . . . 5 ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
3429, 33syl 17 . . . 4 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
35 eldifsn 4292 . . . 4 (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
3625, 34, 35sylanbrc 697 . . 3 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
37 simpl 473 . . 3 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ)
38 cnfldinv 19691 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
3925, 34, 38syl2anc 692 . . . . 5 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
40 zre 11326 . . . . . . . . 9 (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4140adantr 481 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℝ)
42 absresq 13971 . . . . . . . 8 (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2))
4341, 42syl 17 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴↑2))
4426oveq1d 6620 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
45 sq1 12895 . . . . . . . 8 (1↑2) = 1
4644, 45syl6eq 2676 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
4725sqvald 12942 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴↑2) = (𝐴 · 𝐴))
4843, 46, 473eqtr3rd 2669 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴 · 𝐴) = 1)
49 1cnd 10001 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℂ)
5049, 25, 25, 34divmuld 10768 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((1 / 𝐴) = 𝐴 ↔ (𝐴 · 𝐴) = 1))
5148, 50mpbird 247 . . . . 5 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = 𝐴)
5239, 51eqtrd 2660 . . . 4 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = 𝐴)
5352, 37eqeltrd 2704 . . 3 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) ∈ ℤ)
54 cnfldbas 19664 . . . . . 6 ℂ = (Base‘ℂfld)
55 cnfld0 19684 . . . . . 6 0 = (0g‘ℂfld)
56 cndrng 19689 . . . . . 6 fld ∈ DivRing
5754, 55, 56drngui 18669 . . . . 5 (ℂ ∖ {0}) = (Unit‘ℂfld)
58 eqid 2626 . . . . 5 (invr‘ℂfld) = (invr‘ℂfld)
5912, 57, 2, 58subrgunit 18714 . . . 4 (ℤ ∈ (SubRing‘ℂfld) → (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ ∧ ((invr‘ℂfld)‘𝐴) ∈ ℤ)))
604, 59ax-mp 5 . . 3 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ ∧ ((invr‘ℂfld)‘𝐴) ∈ ℤ))
6136, 37, 53, 60syl3anbrc 1244 . 2 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘ℤring))
6223, 61impbii 199 1 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  cdif 3557  wss 3560  {csn 4153  cfv 5850  (class class class)co 6605  cc 9879  cr 9880  0cc0 9881  1c1 9882   · cmul 9886   / cdiv 10629  2c2 11015  cz 11322  cexp 12797  abscabs 13903  ℤ[i]cgz 15552  s cress 15777  Unitcui 18555  invrcinvr 18587  SubRingcsubrg 18692  fldccnfld 19660  ringzring 19732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959  ax-addf 9960  ax-mulf 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12266  df-seq 12739  df-exp 12798  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-gz 15553  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-starv 15872  df-tset 15876  df-ple 15877  df-ds 15880  df-unif 15881  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-subg 17507  df-cmn 18111  df-mgp 18406  df-ur 18418  df-ring 18465  df-cring 18466  df-oppr 18539  df-dvdsr 18557  df-unit 18558  df-invr 18588  df-dvr 18599  df-drng 18665  df-subrg 18694  df-cnfld 19661  df-zring 19733
This theorem is referenced by:  zringndrg  19752  prmirredlem  19755  qqhval2lem  29799
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