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Theorem gzrngunitlem 21369

Description: Lemma for gzrngunit 21370. (Contributed by Mario Carneiro, 4-Dec-2014.)

**Hypothesis**
Ref	Expression
gzrng.1	⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])

**Assertion**
Ref	Expression
gzrngunitlem	⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))

**Proof of Theorem gzrngunitlem**
Step	Hyp	Ref	Expression
1		sq1 14102	. . 3 ⊢ (1↑2) = 1
2		ax-1ne0 11075	. . . . . 6 ⊢ 1 ≠ 0
3		gzsubrg 21358	. . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
4		gzrng.1	. . . . . . . 8 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
5	4	subrgring 20489	. . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
6		eqid 2731	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
7		subrgsubg 20492	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] ∈ (SubGrp‘ℂ_fld))
8		cnfld0 21329	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
9	4, 8	subg0 19045	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂ_fld) → 0 = (0_g‘𝑍))
10	3, 7, 9	mp2b 10	. . . . . . . 8 ⊢ 0 = (0_g‘𝑍)
11		cnfld1 21330	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
12	4, 11	subrg1 20497	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 1 = (1_r‘𝑍))
13	3, 12	ax-mp 5	. . . . . . . 8 ⊢ 1 = (1_r‘𝑍)
14	6, 10, 13	0unit 20314	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0))
15	3, 5, 14	mp2b 10	. . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0)
16	2, 15	nemtbir 3024	. . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍)
17	4	subrgbas 20496	. . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] = (Base‘𝑍))
18	3, 17	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ[i] = (Base‘𝑍)
19	18, 6	unitcl 20293	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
20		gzabssqcl 16853	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ₀)
22		elnn0 12383	. . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ₀ ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0))
23	21, 22	sylib 218	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0))
24	23	ord 864	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0))
25		gzcn 16844	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
26	19, 25	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
27	26	abscld 15346	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
28	27	recnd 11140	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ)
29		sqeq0 14027	. . . . . . . 8 ⊢ ((abs‘𝐴) ∈ ℂ → (((abs‘𝐴)↑2) = 0 ↔ (abs‘𝐴) = 0))
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) = 0 ↔ (abs‘𝐴) = 0))
31	26	abs00ad 15197	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
32		eleq1 2819	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍)))
33	32	biimpcd 249	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍)))
34	31, 33	sylbid 240	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍)))
35	30, 34	sylbid 240	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) = 0 → 0 ∈ (Unit‘𝑍)))
36	24, 35	syld 47	. . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → 0 ∈ (Unit‘𝑍)))
37	16, 36	mt3i 149	. . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ)
38	37	nnge1d 12173	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2))
39	1, 38	eqbrtrid 5124	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2))
40	26	absge0d 15354	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴))
41		1re 11112	. . . 4 ⊢ 1 ∈ ℝ
42		0le1 11640	. . . 4 ⊢ 0 ≤ 1
43		le2sq 14041	. . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
44	41, 42, 43	mpanl12 702	. . 3 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
45	27, 40, 44	syl2anc 584	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
46	39, 45	mpbird 257	1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 ≤ cle 11147 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ↑cexp 13968 abscabs 15141 ℤ[i]cgz 16841 Basecbs 17120 ↾_s cress 17141 0_gc0g 17343 SubGrpcsubg 19033 1_rcur 20099 Ringcrg 20151 Unitcui 20273 SubRingcsubrg 20484 ℂ_fldccnfld 21291

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-gz 16842 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-minusg 18850 df-subg 19036 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-subrng 20461 df-subrg 20485 df-cnfld 21292

This theorem is referenced by: gzrngunit 21370

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