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Mirrors > Home > MPE Home > Th. List > gzrngunitlem | Structured version Visualization version GIF version |
Description: Lemma for gzrngunit 19812. (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 12958 | . . 3 ⊢ (1↑2) = 1 | |
2 | ax-1ne0 10005 | . . . . . 6 ⊢ 1 ≠ 0 | |
3 | gzsubrg 19800 | . . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂfld) | |
4 | gzrng.1 | . . . . . . . 8 ⊢ 𝑍 = (ℂfld ↾s ℤ[i]) | |
5 | 4 | subrgring 18783 | . . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring) |
6 | eqid 2622 | . . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
7 | subrgsubg 18786 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] ∈ (SubGrp‘ℂfld)) | |
8 | cnfld0 19770 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
9 | 4, 8 | subg0 17600 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂfld) → 0 = (0g‘𝑍)) |
10 | 3, 7, 9 | mp2b 10 | . . . . . . . 8 ⊢ 0 = (0g‘𝑍) |
11 | cnfld1 19771 | . . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld) | |
12 | 4, 11 | subrg1 18790 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 1 = (1r‘𝑍)) |
13 | 3, 12 | ax-mp 5 | . . . . . . . 8 ⊢ 1 = (1r‘𝑍) |
14 | 6, 10, 13 | 0unit 18680 | . . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0)) |
15 | 3, 5, 14 | mp2b 10 | . . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0) |
16 | 2, 15 | nemtbir 2889 | . . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍) |
17 | 4 | subrgbas 18789 | . . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍)) |
18 | 3, 17 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ[i] = (Base‘𝑍) |
19 | 18, 6 | unitcl 18659 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i]) |
20 | gzabssqcl 15645 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ0) | |
21 | 19, 20 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ0) |
22 | elnn0 11294 | . . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ0 ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) | |
23 | 21, 22 | sylib 208 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) |
24 | 23 | ord 392 | . . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0)) |
25 | gzcn 15636 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) | |
26 | 19, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ) |
27 | 26 | abscld 14175 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ) |
28 | 27 | recnd 10068 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ) |
29 | sqeq0 12927 | . . . . . . . 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 14030 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) |
32 | eleq1 2689 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍))) | |
33 | 32 | biimpcd 239 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍))) |
34 | 31, 33 | sylbid 230 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍))) |
35 | 30, 34 | sylbid 230 | . . . . . 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 141 | . . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ) |
38 | 37 | nnge1d 11063 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2)) |
39 | 1, 38 | syl5eqbr 4688 | . 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2)) |
40 | 26 | absge0d 14183 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴)) |
41 | 1re 10039 | . . . 4 ⊢ 1 ∈ ℝ | |
42 | 0le1 10551 | . . . 4 ⊢ 0 ≤ 1 | |
43 | le2sq 12938 | . . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) | |
44 | 41, 42, 43 | mpanl12 718 | . . 3 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) |
45 | 27, 40, 44 | syl2anc 693 | . 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) |
46 | 39, 45 | mpbird 247 | 1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 ≤ cle 10075 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ↑cexp 12860 abscabs 13974 ℤ[i]cgz 15633 Basecbs 15857 ↾s cress 15858 0gc0g 16100 SubGrpcsubg 17588 1rcur 18501 Ringcrg 18547 Unitcui 18639 SubRingcsubrg 18776 ℂfldccnfld 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-gz 15634 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-subrg 18778 df-cnfld 19747 |
This theorem is referenced by: gzrngunit 19812 |
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