idlss - Mathbox for Jeff Madsen

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Theorem idlss 38522

Description: Obsolete theorem, use 2idlss 21360 instead. An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

**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 2765	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		idlss.2	. . . 4 ⊢ 𝑋 = ran 𝐺
4		eqid 2765	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 38520	. . 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 481	. 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 1158	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ⊆ wss 3907 ran crn 5652 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 GIdcgi 30747 RingOpscrngo 38400 Idlcidl 38513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6481 df-fun 6527 df-fv 6533 df-ov 7403 df-idl 38516

This theorem is referenced by: idlcl 38523 idlnegcl 38528 1idl 38532 divrngidl 38534 intidl 38535 unichnidl 38537 ispridl2 38544 igenmin 38570 igenidl2 38571

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