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Mirrors > Home > MPE Home > Th. List > gzrngunitlem | Structured version Visualization version GIF version |
Description: Lemma for gzrngunit 20539. (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 13546 | . . 3 ⊢ (1↑2) = 1 | |
2 | ax-1ne0 10594 | . . . . . 6 ⊢ 1 ≠ 0 | |
3 | gzsubrg 20527 | . . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂfld) | |
4 | gzrng.1 | . . . . . . . 8 ⊢ 𝑍 = (ℂfld ↾s ℤ[i]) | |
5 | 4 | subrgring 19467 | . . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring) |
6 | eqid 2818 | . . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
7 | subrgsubg 19470 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] ∈ (SubGrp‘ℂfld)) | |
8 | cnfld0 20497 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
9 | 4, 8 | subg0 18223 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂfld) → 0 = (0g‘𝑍)) |
10 | 3, 7, 9 | mp2b 10 | . . . . . . . 8 ⊢ 0 = (0g‘𝑍) |
11 | cnfld1 20498 | . . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld) | |
12 | 4, 11 | subrg1 19474 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 1 = (1r‘𝑍)) |
13 | 3, 12 | ax-mp 5 | . . . . . . . 8 ⊢ 1 = (1r‘𝑍) |
14 | 6, 10, 13 | 0unit 19359 | . . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0)) |
15 | 3, 5, 14 | mp2b 10 | . . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0) |
16 | 2, 15 | nemtbir 3109 | . . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍) |
17 | 4 | subrgbas 19473 | . . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍)) |
18 | 3, 17 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ[i] = (Base‘𝑍) |
19 | 18, 6 | unitcl 19338 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i]) |
20 | gzabssqcl 16265 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ0) | |
21 | 19, 20 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ0) |
22 | elnn0 11887 | . . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ0 ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) | |
23 | 21, 22 | sylib 219 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) |
24 | 23 | ord 858 | . . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0)) |
25 | gzcn 16256 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) | |
26 | 19, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ) |
27 | 26 | abscld 14784 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ) |
28 | 27 | recnd 10657 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ) |
29 | sqeq0 13474 | . . . . . . . 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 14638 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) |
32 | eleq1 2897 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍))) | |
33 | 32 | biimpcd 250 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍))) |
34 | 31, 33 | sylbid 241 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍))) |
35 | 30, 34 | sylbid 241 | . . . . . 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 151 | . . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ) |
38 | 37 | nnge1d 11673 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2)) |
39 | 1, 38 | eqbrtrid 5092 | . 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2)) |
40 | 26 | absge0d 14792 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴)) |
41 | 1re 10629 | . . . 4 ⊢ 1 ∈ ℝ | |
42 | 0le1 11151 | . . . 4 ⊢ 0 ≤ 1 | |
43 | le2sq 13487 | . . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) | |
44 | 41, 42, 43 | mpanl12 698 | . . 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 258 | 1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 ≤ cle 10664 ℕcn 11626 2c2 11680 ℕ0cn0 11885 ↑cexp 13417 abscabs 14581 ℤ[i]cgz 16253 Basecbs 16471 ↾s cress 16472 0gc0g 16701 SubGrpcsubg 18211 1rcur 19180 Ringcrg 19226 Unitcui 19318 SubRingcsubrg 19460 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-gz 16254 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-subrg 19462 df-cnfld 20474 |
This theorem is referenced by: gzrngunit 20539 |
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