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Mirrors > Home > MPE Home > Th. List > 2idlridld | Structured version Visualization version GIF version |
Description: A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
2idllidld.1 | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
2idlridld.o | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
2idlridld | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2idllidld.1 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
2 | eqid 2732 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | 2idlridld.o | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2732 | . . . 4 ⊢ (LIdeal‘𝑂) = (LIdeal‘𝑂) | |
5 | eqid 2732 | . . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
6 | 2, 3, 4, 5 | 2idlval 20806 | . . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)) |
7 | 1, 6 | eleqtrdi 2843 | . 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))) |
8 | 7 | elin2d 4196 | 1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3944 ‘cfv 6533 opprcoppr 20103 LIdealclidl 20734 2Idealc2idl 20804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-2idl 20805 |
This theorem is referenced by: qsdrng 32521 |
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