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Mirrors > Home > MPE Home > Th. List > gzrngunitlem | Structured version Visualization version GIF version |
Description: Lemma for gzrngunit 19631. (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 12820 | . . 3 ⊢ (1↑2) = 1 | |
2 | ax-1ne0 9884 | . . . . . 6 ⊢ 1 ≠ 0 | |
3 | gzsubrg 19619 | . . . . . . 7 ⊢ ℤ[i] ∈ (SubRing‘ℂfld) | |
4 | gzrng.1 | . . . . . . . 8 ⊢ 𝑍 = (ℂfld ↾s ℤ[i]) | |
5 | 4 | subrgring 18606 | . . . . . . 7 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring) |
6 | eqid 2610 | . . . . . . . 8 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
7 | subrgsubg 18609 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] ∈ (SubGrp‘ℂfld)) | |
8 | cnfld0 19589 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
9 | 4, 8 | subg0 17423 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubGrp‘ℂfld) → 0 = (0g‘𝑍)) |
10 | 3, 7, 9 | mp2b 10 | . . . . . . . 8 ⊢ 0 = (0g‘𝑍) |
11 | cnfld1 19590 | . . . . . . . . . 10 ⊢ 1 = (1r‘ℂfld) | |
12 | 4, 11 | subrg1 18613 | . . . . . . . . 9 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → 1 = (1r‘𝑍)) |
13 | 3, 12 | ax-mp 5 | . . . . . . . 8 ⊢ 1 = (1r‘𝑍) |
14 | 6, 10, 13 | 0unit 18503 | . . . . . . 7 ⊢ (𝑍 ∈ Ring → (0 ∈ (Unit‘𝑍) ↔ 1 = 0)) |
15 | 3, 5, 14 | mp2b 10 | . . . . . 6 ⊢ (0 ∈ (Unit‘𝑍) ↔ 1 = 0) |
16 | 2, 15 | nemtbir 2877 | . . . . 5 ⊢ ¬ 0 ∈ (Unit‘𝑍) |
17 | 4 | subrgbas 18612 | . . . . . . . . . . 11 ⊢ (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍)) |
18 | 3, 17 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ[i] = (Base‘𝑍) |
19 | 18, 6 | unitcl 18482 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i]) |
20 | gzabssqcl 15483 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ0) | |
21 | 19, 20 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ0) |
22 | elnn0 11171 | . . . . . . . 8 ⊢ (((abs‘𝐴)↑2) ∈ ℕ0 ↔ (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) | |
23 | 21, 22 | sylib 207 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) ∈ ℕ ∨ ((abs‘𝐴)↑2) = 0)) |
24 | 23 | ord 391 | . . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → ((abs‘𝐴)↑2) = 0)) |
25 | gzcn 15474 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) | |
26 | 19, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ) |
27 | 26 | abscld 14023 | . . . . . . . . 9 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ) |
28 | 27 | recnd 9947 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℂ) |
29 | sqeq0 12789 | . . . . . . . 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 13878 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) |
32 | eleq1 2676 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 ∈ (Unit‘𝑍) ↔ 0 ∈ (Unit‘𝑍))) | |
33 | 32 | biimpcd 238 | . . . . . . . 8 ⊢ (𝐴 ∈ (Unit‘𝑍) → (𝐴 = 0 → 0 ∈ (Unit‘𝑍))) |
34 | 31, 33 | sylbid 229 | . . . . . . 7 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 → 0 ∈ (Unit‘𝑍))) |
35 | 30, 34 | sylbid 229 | . . . . . 6 ⊢ (𝐴 ∈ (Unit‘𝑍) → (((abs‘𝐴)↑2) = 0 → 0 ∈ (Unit‘𝑍))) |
36 | 24, 35 | syld 46 | . . . . 5 ⊢ (𝐴 ∈ (Unit‘𝑍) → (¬ ((abs‘𝐴)↑2) ∈ ℕ → 0 ∈ (Unit‘𝑍))) |
37 | 16, 36 | mt3i 140 | . . . 4 ⊢ (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴)↑2) ∈ ℕ) |
38 | 37 | nnge1d 10940 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘𝐴)↑2)) |
39 | 1, 38 | syl5eqbr 4618 | . 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1↑2) ≤ ((abs‘𝐴)↑2)) |
40 | 26 | absge0d 14031 | . . 3 ⊢ (𝐴 ∈ (Unit‘𝑍) → 0 ≤ (abs‘𝐴)) |
41 | 1re 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
42 | 0le1 10430 | . . . 4 ⊢ 0 ≤ 1 | |
43 | le2sq 12800 | . . . 4 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) | |
44 | 41, 42, 43 | mpanl12 714 | . . 3 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) |
45 | 27, 40, 44 | syl2anc 691 | . 2 ⊢ (𝐴 ∈ (Unit‘𝑍) → (1 ≤ (abs‘𝐴) ↔ (1↑2) ≤ ((abs‘𝐴)↑2))) |
46 | 39, 45 | mpbird 246 | 1 ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ≤ cle 9954 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 abscabs 13822 ℤ[i]cgz 15471 Basecbs 15695 ↾s cress 15696 0gc0g 15923 SubGrpcsubg 17411 1rcur 18324 Ringcrg 18370 Unitcui 18462 SubRingcsubrg 18599 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-gz 15472 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-subrg 18601 df-cnfld 19568 |
This theorem is referenced by: gzrngunit 19631 |
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