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Theorem gzrngunitlem 21009

Description: Lemma for gzrngunit 21010. (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 14158	. . 3 ⊢ (1↑2) = 1
2		ax-1ne0 11178	. . . . . 6 ⊢ 1 ≠ 0
3		gzsubrg 20998	. . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
4		gzrng.1	. . . . . . . 8 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
5	4	subrgring 20321	. . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
6		eqid 2732	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
7		subrgsubg 20324	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] ∈ (SubGrp‘ℂ_fld))
8		cnfld0 20968	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
9	4, 8	subg0 19011	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂ_fld) → 0 = (0_g‘𝑍))
10	3, 7, 9	mp2b 10	. . . . . . . 8 ⊢ 0 = (0_g‘𝑍)
11		cnfld1 20969	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
12	4, 11	subrg1 20328	. . . . . . . . 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 20209	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0))
15	3, 5, 14	mp2b 10	. . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0)
16	2, 15	nemtbir 3038	. . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍)
17	4	subrgbas 20327	. . . . . . . . . . 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 20188	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
20		gzabssqcl 16873	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ₀)
22		elnn0 12473	. . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ₀ ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0))
23	21, 22	sylib 217	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0))
24	23	ord 862	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0))
25		gzcn 16864	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
26	19, 25	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
27	26	abscld 15382	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
28	27	recnd 11241	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ)
29		sqeq0 14084	. . . . . . . 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 15236	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
32		eleq1 2821	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍)))
33	32	biimpcd 248	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍)))
34	31, 33	sylbid 239	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍)))
35	30, 34	sylbid 239	. . . . . 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 12259	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2))
39	1, 38	eqbrtrid 5183	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2))
40	26	absge0d 15390	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴))
41		1re 11213	. . . 4 ⊢ 1 ∈ ℝ
42		0le1 11736	. . . 4 ⊢ 0 ≤ 1
43		le2sq 14098	. . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
44	41, 42, 43	mpanl12 700	. . 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 256	1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 ≤ cle 11248 ℕcn 12211 2c2 12266 ℕ₀cn0 12471 ↑cexp 14026 abscabs 15180 ℤ[i]cgz 16861 Basecbs 17143 ↾_s cress 17172 0_gc0g 17384 SubGrpcsubg 18999 1_rcur 20003 Ringcrg 20055 Unitcui 20168 SubRingcsubrg 20314 ℂ_fldccnfld 20943

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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12974 df-fz 13484 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-gz 16862 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-subg 19002 df-cmn 19649 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-subrg 20316 df-cnfld 20944

This theorem is referenced by: gzrngunit 21010

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