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Mirrors > Home > MPE Home > Th. List > drngunz | Structured version Visualization version GIF version |
Description: A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.) |
Ref | Expression |
---|---|
drngunz.z | ⊢ 0 = (0g‘𝑅) |
drngunz.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
drngunz | ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 20145 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2738 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | drngunz.u | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | 1unit 20040 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Unit‘𝑅)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → 1 ∈ (Unit‘𝑅)) |
6 | eqid 2738 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | drngunz.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 2, 7 | drngunit 20143 | . . 3 ⊢ (𝑅 ∈ DivRing → ( 1 ∈ (Unit‘𝑅) ↔ ( 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 ))) |
9 | 5, 8 | mpbid 231 | . 2 ⊢ (𝑅 ∈ DivRing → ( 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 )) |
10 | 9 | simprd 497 | 1 ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ‘cfv 6494 Basecbs 17043 0gc0g 17281 1rcur 19872 Ringcrg 19918 Unitcui 20021 DivRingcdr 20138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-2nd 7915 df-tpos 8150 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-plusg 17106 df-mulr 17107 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-drng 20140 |
This theorem is referenced by: abv1 20245 lspsneq 20536 islbs2 20568 islbs3 20569 drngnzr 20685 obsne0 21084 cphsubrglem 24493 ofldlt1 31932 0nellinds 31983 rgmoddim 32121 drngdimgt0 32129 lkrshp 37499 lcfl7lem 39894 fldhmf1 40479 |
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