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| Mirrors > Home > MPE Home > Th. List > Mathboxes > idlss | Structured version Visualization version GIF version | ||
| Description: An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) | 
| Ref | Expression | 
|---|---|
| idlss.1 | ⊢ 𝐺 = (1st ‘𝑅) | 
| idlss.2 | ⊢ 𝑋 = ran 𝐺 | 
| Ref | Expression | 
|---|---|
| idlss | ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idlss.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | eqid 2737 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
| 3 | idlss.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
| 4 | eqid 2737 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
| 5 | 1, 2, 3, 4 | isidl 38021 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼))))) | 
| 6 | 5 | biimpa 476 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)))) | 
| 7 | 6 | simp1d 1143 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 GIdcgi 30509 RingOpscrngo 37901 Idlcidl 38014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-idl 38017 | 
| This theorem is referenced by: idlcl 38024 idlnegcl 38029 1idl 38033 divrngidl 38035 intidl 38036 unichnidl 38038 ispridl2 38045 igenmin 38071 igenidl2 38072 | 
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