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| Mirrors > Home > MPE Home > Th. List > 2idllidld | Structured version Visualization version GIF version | ||
| Description: A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| 2idllidld.1 | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | 
| Ref | Expression | 
|---|---|
| 2idllidld | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 2idllidld.1 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
| 2 | eqid 2736 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
| 3 | eqid 2736 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
| 4 | eqid 2736 | . . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
| 5 | eqid 2736 | . . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
| 6 | 2, 3, 4, 5 | 2idlval 21262 | . . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) | 
| 7 | 1, 6 | eleqtrdi 2850 | . 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))) | 
| 8 | 7 | elin1d 4203 | 1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3949 ‘cfv 6560 opprcoppr 20334 LIdealclidl 21217 2Idealc2idl 21260 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-2idl 21261 | 
| This theorem is referenced by: df2idl2 21268 2idlss 21273 qusmul2idl 21290 rng2idl1cntr 21316 qsnzr 33484 opprqusmulr 33520 opprqus1r 33521 opprqusdrng 33522 qsdrnglem2 33525 qsdrng 33526 | 
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