idlss - Mathbox for Jeff Madsen

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Theorem idlss 38017

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 2730	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		idlss.2	. . . 4 ⊢ 𝑋 = ran 𝐺
4		eqid 2730	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 38015	. . 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 476	. 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ⊆ wss 3917 ran crn 5642 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 GIdcgi 30426 RingOpscrngo 37895 Idlcidl 38008

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-idl 38011

This theorem is referenced by: idlcl 38018 idlnegcl 38023 1idl 38027 divrngidl 38029 intidl 38030 unichnidl 38032 ispridl2 38039 igenmin 38065 igenidl2 38066

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