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Mirrors > Home > MPE Home > Th. List > drngunz | Structured version Visualization version GIF version |
Description: A division ring's unit 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 18802 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2651 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | drngunz.u | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | 1unit 18704 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Unit‘𝑅)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → 1 ∈ (Unit‘𝑅)) |
6 | eqid 2651 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | drngunz.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 2, 7 | drngunit 18800 | . . 3 ⊢ (𝑅 ∈ DivRing → ( 1 ∈ (Unit‘𝑅) ↔ ( 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 ))) |
9 | 5, 8 | mpbid 222 | . 2 ⊢ (𝑅 ∈ DivRing → ( 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 )) |
10 | 9 | simprd 478 | 1 ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ‘cfv 5926 Basecbs 15904 0gc0g 16147 1rcur 18547 Ringcrg 18593 Unitcui 18685 DivRingcdr 18795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-mulr 16002 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-drng 18797 |
This theorem is referenced by: abv1 18881 lspsneq 19170 islbs2 19202 islbs3 19203 drngnzr 19310 obsne0 20117 cphsubrglem 23023 ofldlt1 29941 lkrshp 34710 lcfl7lem 37105 |
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