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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 21390. (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 21369 | . 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽) |
| 6 | 5 | elin2 4155 | 1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 209 ∧ wa 400 = wceq 1568 ∈ wcel 2141 ‘cfv 6536 opprcoppr 20417 LIdealclidl 21309 2Idealc2idl 21367 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pr 5404 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-iota 6492 df-fun 6538 df-fv 6544 df-2idl 21368 |
| This theorem is referenced by: df2idl2rng 21374 2idlelbas 21382 rng2idlsubgsubrng 21386 2idlcpblrng 21389 2idlcpbl 21390 |
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