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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 2724 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | idlss.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2724 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 1, 2, 3, 4 | isidl 37385 | . . 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 1139 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3941 ran crn 5668 ‘cfv 6534 (class class class)co 7402 1st c1st 7967 2nd c2nd 7968 GIdcgi 30237 RingOpscrngo 37265 Idlcidl 37378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-idl 37381 |
This theorem is referenced by: idlcl 37388 idlnegcl 37393 1idl 37397 divrngidl 37399 intidl 37400 unichnidl 37402 ispridl2 37409 igenmin 37435 igenidl2 37436 |
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