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Mirrors > Home > MPE Home > Th. List > crngridl | Structured version Visualization version GIF version |
Description: In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
crng2idl.i | ⊢ 𝐼 = (LIdeal‘𝑅) |
crngridl.o | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
crngridl | ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crng2idl.i | . 2 ⊢ 𝐼 = (LIdeal‘𝑅) | |
2 | eqidd 2826 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅)) | |
3 | crngridl.o | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2825 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 3, 4 | opprbas 18990 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑂)) |
7 | ssv 3850 | . . . . 5 ⊢ (Base‘𝑅) ⊆ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ⊆ V) |
9 | eqid 2825 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
10 | 3, 9 | oppradd 18991 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑂) |
11 | 10 | oveqi 6923 | . . . . 5 ⊢ (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦) |
12 | 11 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦)) |
13 | ovexd 6944 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ V) | |
14 | eqid 2825 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
15 | eqid 2825 | . . . . . 6 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
16 | 4, 14, 3, 15 | crngoppr 18988 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑂)𝑦)) |
17 | 16 | 3expb 1153 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑂)𝑦)) |
18 | 2, 6, 8, 12, 13, 17 | lidlrsppropd 19598 | . . 3 ⊢ (𝑅 ∈ CRing → ((LIdeal‘𝑅) = (LIdeal‘𝑂) ∧ (RSpan‘𝑅) = (RSpan‘𝑂))) |
19 | 18 | simpld 490 | . 2 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (LIdeal‘𝑂)) |
20 | 1, 19 | syl5eq 2873 | 1 ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ⊆ wss 3798 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 CRingccrg 18909 opprcoppr 18983 LIdealclidl 19538 RSpancrsp 19539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-cmn 18555 df-mgp 18851 df-cring 18911 df-oppr 18984 df-lss 19296 df-lsp 19338 df-sra 19540 df-rgmod 19541 df-lidl 19542 df-rsp 19543 |
This theorem is referenced by: crng2idl 19607 |
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