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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 2822 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅)) | |
3 | crngridl.o | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 3, 4 | opprbas 19373 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑂)) |
7 | ssv 3990 | . . . . 5 ⊢ (Base‘𝑅) ⊆ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ⊆ V) |
9 | eqid 2821 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
10 | 3, 9 | oppradd 19374 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑂) |
11 | 10 | oveqi 7163 | . . . . 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 7185 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ V) | |
14 | eqid 2821 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
15 | eqid 2821 | . . . . . 6 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
16 | 4, 14, 3, 15 | crngoppr 19371 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑂)𝑦)) |
17 | 16 | 3expb 1116 | . . . 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 19997 | . . 3 ⊢ (𝑅 ∈ CRing → ((LIdeal‘𝑅) = (LIdeal‘𝑂) ∧ (RSpan‘𝑅) = (RSpan‘𝑂))) |
19 | 18 | simpld 497 | . 2 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (LIdeal‘𝑂)) |
20 | 1, 19 | syl5eq 2868 | 1 ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 CRingccrg 19292 opprcoppr 19366 LIdealclidl 19936 RSpancrsp 19937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-cmn 18902 df-mgp 19234 df-cring 19294 df-oppr 19367 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-lidl 19940 df-rsp 19941 |
This theorem is referenced by: crng2idl 20006 |
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