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| Mirrors > Home > MPE Home > Th. List > 2idlelb | Structured version Visualization version GIF version | ||
| Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21283. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| 2idlel.i | ⊢ 𝐼 = (LIdeal‘𝑅) | 
| 2idlel.o | ⊢ 𝑂 = (oppr‘𝑅) | 
| 2idlel.j | ⊢ 𝐽 = (LIdeal‘𝑂) | 
| 2idlel.t | ⊢ 𝑇 = (2Ideal‘𝑅) | 
| Ref | Expression | 
|---|---|
| 2idlelb | ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 2idlel.i | . . 3 ⊢ 𝐼 = (LIdeal‘𝑅) | |
| 2 | 2idlel.o | . . 3 ⊢ 𝑂 = (oppr‘𝑅) | |
| 3 | 2idlel.j | . . 3 ⊢ 𝐽 = (LIdeal‘𝑂) | |
| 4 | 2idlel.t | . . 3 ⊢ 𝑇 = (2Ideal‘𝑅) | |
| 5 | 1, 2, 3, 4 | 2idlval 21262 | . 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽) | 
| 6 | 5 | elin2 4202 | 1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘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: df2idl2rng 21267 2idlelbas 21275 rng2idlsubgsubrng 21279 2idlcpblrng 21282 2idlcpbl 21283 | 
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