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Mirrors > Home > MPE Home > Th. List > zringndrg | Structured version Visualization version GIF version |
Description: The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.) |
Ref | Expression |
---|---|
zringndrg | ⊢ ℤring ∉ DivRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ne2 11240 | . . . . . . 7 ⊢ 1 ≠ 2 | |
2 | 1 | nesymi 2851 | . . . . . 6 ⊢ ¬ 2 = 1 |
3 | 2re 11090 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
4 | 0le2 11111 | . . . . . . . 8 ⊢ 0 ≤ 2 | |
5 | absid 14036 | . . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
6 | 3, 4, 5 | mp2an 708 | . . . . . . 7 ⊢ (abs‘2) = 2 |
7 | 6 | eqeq1i 2627 | . . . . . 6 ⊢ ((abs‘2) = 1 ↔ 2 = 1) |
8 | 2, 7 | mtbir 313 | . . . . 5 ⊢ ¬ (abs‘2) = 1 |
9 | 8 | intnan 960 | . . . 4 ⊢ ¬ (2 ∈ ℤ ∧ (abs‘2) = 1) |
10 | zringunit 19836 | . . . 4 ⊢ (2 ∈ (Unit‘ℤring) ↔ (2 ∈ ℤ ∧ (abs‘2) = 1)) | |
11 | 9, 10 | mtbir 313 | . . 3 ⊢ ¬ 2 ∈ (Unit‘ℤring) |
12 | zringbas 19824 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
13 | eqid 2622 | . . . . 5 ⊢ (Unit‘ℤring) = (Unit‘ℤring) | |
14 | zring0 19828 | . . . . 5 ⊢ 0 = (0g‘ℤring) | |
15 | 12, 13, 14 | isdrng 18751 | . . . 4 ⊢ (ℤring ∈ DivRing ↔ (ℤring ∈ Ring ∧ (Unit‘ℤring) = (ℤ ∖ {0}))) |
16 | 2z 11409 | . . . . . 6 ⊢ 2 ∈ ℤ | |
17 | 2ne0 11113 | . . . . . 6 ⊢ 2 ≠ 0 | |
18 | eldifsn 4317 | . . . . . 6 ⊢ (2 ∈ (ℤ ∖ {0}) ↔ (2 ∈ ℤ ∧ 2 ≠ 0)) | |
19 | 16, 17, 18 | mpbir2an 955 | . . . . 5 ⊢ 2 ∈ (ℤ ∖ {0}) |
20 | id 22 | . . . . 5 ⊢ ((Unit‘ℤring) = (ℤ ∖ {0}) → (Unit‘ℤring) = (ℤ ∖ {0})) | |
21 | 19, 20 | syl5eleqr 2708 | . . . 4 ⊢ ((Unit‘ℤring) = (ℤ ∖ {0}) → 2 ∈ (Unit‘ℤring)) |
22 | 15, 21 | simplbiim 659 | . . 3 ⊢ (ℤring ∈ DivRing → 2 ∈ (Unit‘ℤring)) |
23 | 11, 22 | mto 188 | . 2 ⊢ ¬ ℤring ∈ DivRing |
24 | 23 | nelir 2900 | 1 ⊢ ℤring ∉ DivRing |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∖ cdif 3571 {csn 4177 class class class wbr 4653 ‘cfv 5888 ℝcr 9935 0cc0 9936 1c1 9937 ≤ cle 10075 2c2 11070 ℤcz 11377 abscabs 13974 Ringcrg 18547 Unitcui 18639 DivRingcdr 18747 ℤringzring 19818 |
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-dvr 18683 df-drng 18749 df-subrg 18778 df-cnfld 19747 df-zring 19819 |
This theorem is referenced by: zclmncvs 22948 |
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