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Theorem gzrngunitlem 21411

Description: Lemma for gzrngunit 21412. (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 14152	. . 3 ⊢ (1↑2) = 1
2		ax-1ne0 11102	. . . . . 6 ⊢ 1 ≠ 0
3		gzsubrg 21400	. . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
4		gzrng.1	. . . . . . . 8 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
5	4	subrgring 20550	. . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
6		eqid 2741	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
7		subrgsubg 20553	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] ∈ (SubGrp‘ℂ_fld))
8		cnfld0 21375	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
9	4, 8	subg0 19103	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂ_fld) → 0 = (0_g‘𝑍))
10	3, 7, 9	mp2b 10	. . . . . . . 8 ⊢ 0 = (0_g‘𝑍)
11		cnfld1 21376	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
12	4, 11	subrg1 20558	. . . . . . . . 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 20371	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0))
15	3, 5, 14	mp2b 10	. . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0)
16	2, 15	nemtbir 3032	. . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍)
17	4	subrgbas 20557	. . . . . . . . . . 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 20350	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
20		gzabssqcl 16907	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ₀)
22		elnn0 12434	. . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ₀ ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0))
23	21, 22	sylib 220	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0))
24	23	ord 871	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0))
25		gzcn 16898	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
26	19, 25	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
27	26	abscld 15396	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
28	27	recnd 11168	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ)
29		sqeq0 14077	. . . . . . . 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 15247	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
32		eleq1 2829	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍)))
33	32	biimpcd 251	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍)))
34	31, 33	sylbid 242	. . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍)))
35	30, 34	sylbid 242	. . . . . 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 12220	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2))
39	1, 38	eqbrtrid 5110	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2))
40	26	absge0d 15404	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴))
41		1re 11139	. . . 4 ⊢ 1 ∈ ℝ
42		0le1 11668	. . . 4 ⊢ 0 ≤ 1
43		le2sq 14091	. . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
44	41, 42, 43	mpanl12 709	. . 3 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
45	27, 40, 44	syl2anc 591	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2)))
46	39, 45	mpbird 259	1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∨ wo 854 = wceq 1548 ∈ wcel 2121 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 ≤ cle 11175 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 ↑cexp 14018 abscabs 15191 ℤ[i]cgz 16895 Basecbs 17174 ↾_s cress 17195 0_gc0g 17397 SubGrpcsubg 19091 1_rcur 20157 Ringcrg 20209 Unitcui 20330 SubRingcsubrg 20545 ℂ_fldccnfld 21351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-gz 16896 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-0g 17399 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-subg 19094 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-subrng 20522 df-subrg 20546 df-cnfld 21352

This theorem is referenced by: gzrngunit 21412

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