idlss - Mathbox for Jeff Madsen

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Theorem idlss 37976

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 2740	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		idlss.2	. . . 4 ⊢ 𝑋 = ran 𝐺
4		eqid 2740	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 37974	. . 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 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ⊆ wss 3976 ran crn 5701 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 GIdcgi 30522 RingOpscrngo 37854 Idlcidl 37967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-ov 7451 df-idl 37970

This theorem is referenced by: idlcl 37977 idlnegcl 37982 1idl 37986 divrngidl 37988 intidl 37989 unichnidl 37991 ispridl2 37998 igenmin 38024 igenidl2 38025

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