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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 2739 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅)) | |
3 | crngridl.o | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 3, 4 | opprbas 19660 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑂)) |
7 | ssv 3934 | . . . . 5 ⊢ (Base‘𝑅) ⊆ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ⊆ V) |
9 | eqid 2738 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
10 | 3, 9 | oppradd 19661 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑂) |
11 | 10 | oveqi 7235 | . . . . 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 7257 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ V) | |
14 | eqid 2738 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
15 | eqid 2738 | . . . . . 6 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
16 | 4, 14, 3, 15 | crngoppr 19658 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑂)𝑦)) |
17 | 16 | 3expb 1122 | . . . 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 20281 | . . 3 ⊢ (𝑅 ∈ CRing → ((LIdeal‘𝑅) = (LIdeal‘𝑂) ∧ (RSpan‘𝑅) = (RSpan‘𝑂))) |
19 | 18 | simpld 498 | . 2 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (LIdeal‘𝑂)) |
20 | 1, 19 | syl5eq 2791 | 1 ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3415 ⊆ wss 3875 ‘cfv 6389 (class class class)co 7222 Basecbs 16773 +gcplusg 16815 .rcmulr 16816 CRingccrg 19576 opprcoppr 19653 LIdealclidl 20220 RSpancrsp 20221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-tpos 7977 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-er 8400 df-en 8636 df-dom 8637 df-sdom 8638 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-nn 11844 df-2 11906 df-3 11907 df-4 11908 df-5 11909 df-6 11910 df-7 11911 df-8 11912 df-sets 16730 df-slot 16748 df-ndx 16758 df-base 16774 df-ress 16798 df-plusg 16828 df-mulr 16829 df-sca 16831 df-vsca 16832 df-ip 16833 df-cmn 19185 df-mgp 19518 df-cring 19578 df-oppr 19654 df-lss 19982 df-lsp 20022 df-sra 20222 df-rgmod 20223 df-lidl 20224 df-rsp 20225 |
This theorem is referenced by: crng2idl 20290 |
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