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| Mirrors > Home > MPE Home > Th. List > gzrngunitlem | Structured version Visualization version GIF version | ||
| Description: Lemma for gzrngunit 21451. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| Ref | Expression |
|---|---|
| gzrng.1 | ⊢ 𝑍 = (ℂfld ↾s ℤ[i]) |
| Ref | Expression |
|---|---|
| gzrngunitlem | ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) |
| 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‘ℂfld) | |
| 4 | gzrng.1 | . . . . . . . 8 ⊢ 𝑍 = (ℂfld ↾s ℤ[i]) | |
| 5 | 4 | subrgring 20574 | . . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring) |
| 6 | eqid 2737 | . . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
| 7 | subrgsubg 20577 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] ∈ (SubGrp‘ℂfld)) | |
| 8 | cnfld0 21405 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
| 9 | 4, 8 | subg0 19150 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂfld) → 0 = (0g‘𝑍)) |
| 10 | 3, 7, 9 | mp2b 10 | . . . . . . . 8 ⊢ 0 = (0g‘𝑍) |
| 11 | cnfld1 21406 | . . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld) | |
| 12 | 4, 11 | subrg1 20582 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 1 = (1r‘𝑍)) |
| 13 | 3, 12 | ax-mp 5 | . . . . . . . 8 ⊢ 1 = (1r‘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‘ℂfld) → 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 0gc0g 17484 SubGrpcsubg 19138 1rcur 20178 Ringcrg 20230 Unitcui 20355 SubRingcsubrg 20569 ℂfldccnfld 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 |
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