gzrngunitlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > gzrngunitlem		Structured version Visualization version GIF version

Theorem gzrngunitlem 21450

Description: Lemma for gzrngunit 21451. (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 14234	. . 3 ⊢ (1↑2) = 1
2		ax-1ne0 11224	. . . . . 6 ⊢ 1 ≠ 0
3		gzsubrg 21439	. . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂ_fld)
4		gzrng.1	. . . . . . . 8 ⊢ 𝑍 = (ℂ_fld ↾_s ℤ[i])
5	4	subrgring 20574	. . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → 𝑍 ∈ Ring)
6		eqid 2737	. . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
7		subrgsubg 20577	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂ_fld) → ℤ[i] ∈ (SubGrp‘ℂ_fld))
8		cnfld0 21405	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
9	4, 8	subg0 19150	. . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂ_fld) → 0 = (0_g‘𝑍))
10	3, 7, 9	mp2b 10	. . . . . . . 8 ⊢ 0 = (0_g‘𝑍)
11		cnfld1 21406	. . . . . . . . . 10 ⊢ 1 = (1_r‘ℂ_fld)
12	4, 11	subrg1 20582	. . . . . . . . 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 20396	. . . . . . 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 20581	. . . . . . . . . . 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 20375	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
20		gzabssqcl 16979	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ₀)
22		elnn0 12528	. . . . . . . 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 865	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0))
25		gzcn 16970	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
26	19, 25	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
27	26	abscld 15475	. . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
28	27	recnd 11289	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ)
29		sqeq0 14160	. . . . . . . 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 15329	. . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
32		eleq1 2829	. . . . . . . . 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 12314	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2))
39	1, 38	eqbrtrid 5178	. 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2))
40	26	absge0d 15483	. . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴))
41		1re 11261	. . . 4 ⊢ 1 ∈ ℝ
42		0le1 11786	. . . 4 ⊢ 0 ≤ 1
43		le2sq 14174	. . . 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 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ≤ cle 11296 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ↑cexp 14102 abscabs 15273 ℤ[i]cgz 16967 Basecbs 17247 ↾_s cress 17274 0_gc0g 17484 SubGrpcsubg 19138 1_rcur 20178 Ringcrg 20230 Unitcui 20355 SubRingcsubrg 20569 ℂ_fldccnfld 21364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-gz 16968 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrng 20546 df-subrg 20570 df-cnfld 21365

This theorem is referenced by: gzrngunit 21451

W3C validator