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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 2735 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | eqid 2735 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
4 | eqid 2735 | . . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
5 | eqid 2735 | . . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
6 | 2, 3, 4, 5 | 2idlval 21279 | . . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
7 | 1, 6 | eleqtrdi 2849 | . 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))) |
8 | 7 | elin1d 4214 | 1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3962 ‘cfv 6563 opprcoppr 20350 LIdealclidl 21234 2Idealc2idl 21277 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-2idl 21278 |
This theorem is referenced by: df2idl2 21285 2idlss 21290 qusmul2idl 21307 rng2idl1cntr 21333 qsnzr 33463 opprqusmulr 33499 opprqus1r 33500 opprqusdrng 33501 qsdrnglem2 33504 qsdrng 33505 |
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