idlss - Mathbox for Jeff Madsen

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Theorem idlss 36879

Description: An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Hypotheses**
Ref	Expression
idlss.1	⊢ 𝐺 = (1^st ‘𝑅)
idlss.2	⊢ 𝑋 = ran 𝐺

**Assertion**
Ref	Expression
idlss	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)

Proof of Theorem idlss Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idlss.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑅)
2		eqid 2732	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		idlss.2	. . . 4 ⊢ 𝑋 = ran 𝐺
4		eqid 2732	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 36877	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2^nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ 𝐼)))))
6	5	biimpa 477	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2^nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ 𝐼))))
7	6	simp1d 1142	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 ran crn 5677 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 GIdcgi 29738 RingOpscrngo 36757 Idlcidl 36870

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-idl 36873

This theorem is referenced by: idlcl 36880 idlnegcl 36885 1idl 36889 divrngidl 36891 intidl 36892 unichnidl 36894 ispridl2 36901 igenmin 36927 igenidl2 36928

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