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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 2733 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | eqid 2733 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
4 | eqid 2733 | . . . 4 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
5 | eqid 2733 | . . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
6 | 2, 3, 4, 5 | 2idlval 20845 | . . 3 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
7 | 1, 6 | eleqtrdi 2844 | . 2 ⊢ (𝜑 → 𝐼 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅)))) |
8 | 7 | elin1d 4197 | 1 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3946 ‘cfv 6540 opprcoppr 20138 LIdealclidl 20771 2Idealc2idl 20843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-2idl 20844 |
This theorem is referenced by: 2idlss 20855 qusmul2 20862 qsnzr 32532 opprqusmulr 32558 opprqus1r 32559 opprqusdrng 32560 qsdrnglem2 32563 qsdrng 32564 rng2idl1cntr 46719 |
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