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Theorem zringunit 19655
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 19643 . . . 4 ℤ = (Base‘ℤring)
2 eqid 2610 . . . 4 (Unit‘ℤring) = (Unit‘ℤring)
31, 2unitcl 18482 . . 3 (𝐴 ∈ (Unit‘ℤring) → 𝐴 ∈ ℤ)
4 zsubrg 19618 . . . . . . 7 ℤ ∈ (SubRing‘ℂfld)
5 zgz 15475 . . . . . . . 8 (𝑥 ∈ ℤ → 𝑥 ∈ ℤ[i])
65ssriv 3572 . . . . . . 7 ℤ ⊆ ℤ[i]
7 gzsubrg 19619 . . . . . . . 8 ℤ[i] ∈ (SubRing‘ℂfld)
8 eqid 2610 . . . . . . . . 9 (ℂflds ℤ[i]) = (ℂflds ℤ[i])
98subsubrg 18629 . . . . . . . 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 957 . . . . . 6 ℤ ∈ (SubRing‘(ℂflds ℤ[i]))
12 df-zring 19638 . . . . . . . 8 ring = (ℂflds ℤ)
13 ressabs 15766 . . . . . . . . 9 ((ℤ[i] ∈ (SubRing‘ℂfld) ∧ ℤ ⊆ ℤ[i]) → ((ℂflds ℤ[i]) ↾s ℤ) = (ℂflds ℤ))
147, 6, 13mp2an 704 . . . . . . . 8 ((ℂflds ℤ[i]) ↾s ℤ) = (ℂflds ℤ)
1512, 14eqtr4i 2635 . . . . . . 7 ring = ((ℂflds ℤ[i]) ↾s ℤ)
16 eqid 2610 . . . . . . 7 (Unit‘(ℂflds ℤ[i])) = (Unit‘(ℂflds ℤ[i]))
1715, 16, 2subrguss 18618 . . . . . 6 (ℤ ∈ (SubRing‘(ℂflds ℤ[i])) → (Unit‘ℤring) ⊆ (Unit‘(ℂflds ℤ[i])))
1811, 17ax-mp 5 . . . . 5 (Unit‘ℤring) ⊆ (Unit‘(ℂflds ℤ[i]))
1918sseli 3564 . . . 4 (𝐴 ∈ (Unit‘ℤring) → 𝐴 ∈ (Unit‘(ℂflds ℤ[i])))
208gzrngunit 19631 . . . . 5 (𝐴 ∈ (Unit‘(ℂflds ℤ[i])) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
2120simprbi 479 . . . 4 (𝐴 ∈ (Unit‘(ℂflds ℤ[i])) → (abs‘𝐴) = 1)
2219, 21syl 17 . . 3 (𝐴 ∈ (Unit‘ℤring) → (abs‘𝐴) = 1)
233, 22jca 553 . 2 (𝐴 ∈ (Unit‘ℤring) → (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
24 zcn 11259 . . . . 5 (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
2524adantr 480 . . . 4 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
26 simpr 476 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
27 ax-1ne0 9884 . . . . . . 7 1 ≠ 0
2827a1i 11 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
2926, 28eqnetrd 2849 . . . . 5 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
30 fveq2 6103 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
31 abs0 13873 . . . . . . 7 (abs‘0) = 0
3230, 31syl6eq 2660 . . . . . 6 (𝐴 = 0 → (abs‘𝐴) = 0)
3332necon3i 2814 . . . . 5 ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
3429, 33syl 17 . . . 4 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
35 eldifsn 4260 . . . 4 (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
3625, 34, 35sylanbrc 695 . . 3 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
37 simpl 472 . . 3 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ)
38 cnfldinv 19596 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
3925, 34, 38syl2anc 691 . . . . 5 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
40 zre 11258 . . . . . . . . 9 (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4140adantr 480 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℝ)
42 absresq 13890 . . . . . . . 8 (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2))
4341, 42syl 17 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴↑2))
4426oveq1d 6564 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
45 sq1 12820 . . . . . . . 8 (1↑2) = 1
4644, 45syl6eq 2660 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
4725sqvald 12867 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴↑2) = (𝐴 · 𝐴))
4843, 46, 473eqtr3rd 2653 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (𝐴 · 𝐴) = 1)
49 1cnd 9935 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℂ)
5049, 25, 25, 34divmuld 10702 . . . . . 6 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((1 / 𝐴) = 𝐴 ↔ (𝐴 · 𝐴) = 1))
5148, 50mpbird 246 . . . . 5 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = 𝐴)
5239, 51eqtrd 2644 . . . 4 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = 𝐴)
5352, 37eqeltrd 2688 . . 3 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) ∈ ℤ)
54 cnfldbas 19571 . . . . . 6 ℂ = (Base‘ℂfld)
55 cnfld0 19589 . . . . . 6 0 = (0g‘ℂfld)
56 cndrng 19594 . . . . . 6 fld ∈ DivRing
5754, 55, 56drngui 18576 . . . . 5 (ℂ ∖ {0}) = (Unit‘ℂfld)
58 eqid 2610 . . . . 5 (invr‘ℂfld) = (invr‘ℂfld)
5912, 57, 2, 58subrgunit 18621 . . . 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 1239 . 2 ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘ℤring))
6223, 61impbii 198 1 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  cdif 3537  wss 3540  {csn 4125  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815  1c1 9816   · cmul 9820   / cdiv 10563  2c2 10947  cz 11254  cexp 12722  abscabs 13822  ℤ[i]cgz 15471  s cress 15696  Unitcui 18462  invrcinvr 18494  SubRingcsubrg 18599  fldccnfld 19567  ringzring 19637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-gz 15472  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-cmn 18018  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-subrg 18601  df-cnfld 19568  df-zring 19638
This theorem is referenced by:  zringndrg  19657  prmirredlem  19660  qqhval2lem  29353
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