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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 2735 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | idlss.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2735 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 1, 2, 3, 4 | isidl 38001 | . . 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 1141 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ran crn 5690 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 GIdcgi 30519 RingOpscrngo 37881 Idlcidl 37994 |
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-pow 5371 ax-pr 5438 ax-un 7754 |
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-pw 4607 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-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-idl 37997 |
This theorem is referenced by: idlcl 38004 idlnegcl 38009 1idl 38013 divrngidl 38015 intidl 38016 unichnidl 38018 ispridl2 38025 igenmin 38051 igenidl2 38052 |
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