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Theorem gzrngunitlem 21467

Description: Lemma for gzrngunit 21468. (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 14230	. . 3 ⊢ (1↑2) = 1
2		ax-1ne0 11221	. . . . . 6 ⊢ 1 ≠ 0
3		gzsubrg 21456	. . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
4		gzrng.1	. . . . . . . 8 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
5	4	subrgring 20590	. . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
6		eqid 2734	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
7		subrgsubg 20593	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] ∈ (SubGrp‘ℂ_fld))
8		cnfld0 21422	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
9	4, 8	subg0 19162	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂ_fld) → 0 = (0_g‘𝑍))
10	3, 7, 9	mp2b 10	. . . . . . . 8 ⊢ 0 = (0_g‘𝑍)
11		cnfld1 21423	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
12	4, 11	subrg1 20598	. . . . . . . . 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 20412	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0))
15	3, 5, 14	mp2b 10	. . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0)
16	2, 15	nemtbir 3035	. . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍)
17	4	subrgbas 20597	. . . . . . . . . . 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 20391	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
20		gzabssqcl 16974	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ₀)
22		elnn0 12525	. . . . . . . 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 16965	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
26	19, 25	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
27	26	abscld 15471	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
28	27	recnd 11286	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ)
29		sqeq0 14156	. . . . . . . 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 15325	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
32		eleq1 2826	. . . . . . . . 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 12311	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2))
39	1, 38	eqbrtrid 5182	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2))
40	26	absge0d 15479	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴))
41		1re 11258	. . . 4 ⊢ 1 ∈ ℝ
42		0le1 11783	. . . 4 ⊢ 0 ≤ 1
43		le2sq 14170	. . . 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 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 ≤ cle 11293 ℕcn 12263 2c2 12318 ℕ₀cn0 12523 ↑cexp 14098 abscabs 15269 ℤ[i]cgz 16962 Basecbs 17244 ↾_s cress 17273 0_gc0g 17485 SubGrpcsubg 19150 1_rcur 20198 Ringcrg 20250 Unitcui 20371 SubRingcsubrg 20585 ℂ_fldccnfld 21381

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-gz 16963 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-subrng 20562 df-subrg 20586 df-cnfld 21382

This theorem is referenced by: gzrngunit 21468

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