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Mirrors > Home > MPE Home > Th. List > gzrngunitlem | Structured version Visualization version GIF version |
Description: Lemma for gzrngunit 20945. (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 14141 | . . 3 ⊢ (1↑2) = 1 | |
2 | ax-1ne0 11161 | . . . . . 6 ⊢ 1 ≠ 0 | |
3 | gzsubrg 20933 | . . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂfld) | |
4 | gzrng.1 | . . . . . . . 8 ⊢ 𝑍 = (ℂfld ↾s ℤ[i]) | |
5 | 4 | subrgring 20315 | . . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring) |
6 | eqid 2731 | . . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
7 | subrgsubg 20318 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] ∈ (SubGrp‘ℂfld)) | |
8 | cnfld0 20903 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
9 | 4, 8 | subg0 18984 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂfld) → 0 = (0g‘𝑍)) |
10 | 3, 7, 9 | mp2b 10 | . . . . . . . 8 ⊢ 0 = (0g‘𝑍) |
11 | cnfld1 20904 | . . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld) | |
12 | 4, 11 | subrg1 20322 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 1 = (1r‘𝑍)) |
13 | 3, 12 | ax-mp 5 | . . . . . . . 8 ⊢ 1 = (1r‘𝑍) |
14 | 6, 10, 13 | 0unit 20162 | . . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0)) |
15 | 3, 5, 14 | mp2b 10 | . . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0) |
16 | 2, 15 | nemtbir 3037 | . . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍) |
17 | 4 | subrgbas 20321 | . . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍)) |
18 | 3, 17 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ[i] = (Base‘𝑍) |
19 | 18, 6 | unitcl 20141 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i]) |
20 | gzabssqcl 16856 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ0) | |
21 | 19, 20 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ0) |
22 | elnn0 12456 | . . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ0 ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) | |
23 | 21, 22 | sylib 217 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) |
24 | 23 | ord 862 | . . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0)) |
25 | gzcn 16847 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) | |
26 | 19, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ) |
27 | 26 | abscld 15365 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ) |
28 | 27 | recnd 11224 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ) |
29 | sqeq0 14067 | . . . . . . . 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 15219 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) |
32 | eleq1 2820 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍))) | |
33 | 32 | biimpcd 248 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍))) |
34 | 31, 33 | sylbid 239 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍))) |
35 | 30, 34 | sylbid 239 | . . . . . 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 12242 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2)) |
39 | 1, 38 | eqbrtrid 5176 | . 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2)) |
40 | 26 | absge0d 15373 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴)) |
41 | 1re 11196 | . . . 4 ⊢ 1 ∈ ℝ | |
42 | 0le1 11719 | . . . 4 ⊢ 0 ≤ 1 | |
43 | le2sq 14081 | . . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) | |
44 | 41, 42, 43 | mpanl12 700 | . . 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 256 | 1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 ℝcr 11091 0cc0 11092 1c1 11093 ≤ cle 11231 ℕcn 12194 2c2 12249 ℕ0cn0 12454 ↑cexp 14009 abscabs 15163 ℤ[i]cgz 16844 Basecbs 17126 ↾s cress 17155 0gc0g 17367 SubGrpcsubg 18972 1rcur 19963 Ringcrg 20014 Unitcui 20121 SubRingcsubrg 20308 ℂfldccnfld 20878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-rp 12957 df-fz 13467 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-gz 16845 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-subg 18975 df-cmn 19614 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-subrg 20310 df-cnfld 20879 |
This theorem is referenced by: gzrngunit 20945 |
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