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Mirrors > Home > MPE Home > Th. List > ringideu | Structured version Visualization version GIF version |
Description: The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
ringcl.b | ⊢ 𝐵 = (Base‘𝑅) |
ringcl.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
ringideu | ⊢ (𝑅 ∈ Ring → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | ringmgp 18760 | . 2 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
3 | ringcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 3 | mgpbas 18702 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
5 | ringcl.t | . . . 4 ⊢ · = (.r‘𝑅) | |
6 | 1, 5 | mgpplusg 18700 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
7 | 4, 6 | mndideu 17511 | . 2 ⊢ ((mulGrp‘𝑅) ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥)) |
8 | 2, 7 | syl 17 | 1 ⊢ (𝑅 ∈ Ring → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃!wreu 3063 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 .rcmulr 16149 Mndcmnd 17501 mulGrpcmgp 18696 Ringcrg 18754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-plusg 16161 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mgp 18697 df-ring 18756 |
This theorem is referenced by: isringid 18780 isringrng 42405 uzlidlring 42453 |
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