idlss - Mathbox for Jeff Madsen

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Theorem idlss 35860

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 2736	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		idlss.2	. . . 4 ⊢ 𝑋 = ran 𝐺
4		eqid 2736	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 35858	. . 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 480	. 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 1144	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 ran crn 5537 ‘cfv 6358 (class class class)co 7191 1^st c1st 7737 2^nd c2nd 7738 GIdcgi 28525 RingOpscrngo 35738 Idlcidl 35851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-idl 35854

This theorem is referenced by: idlcl 35861 idlnegcl 35866 1idl 35870 divrngidl 35872 intidl 35873 unichnidl 35875 ispridl2 35882 igenmin 35908 igenidl2 35909

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