Theorem isidlc 38022

Description: The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
idlval.1	⊢ 𝐺 = (1st ‘𝑅)
idlval.2	⊢ 𝐻 = (2nd ‘𝑅)
idlval.3	⊢ 𝑋 = ran 𝐺
idlval.4	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
isidlc	⊢ (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))

Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑧,𝑋   𝑥,𝐼,𝑦,𝑧   𝑥,𝑋

Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝑋(𝑦)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem isidlc

Step	Hyp	Ref	Expression
1		crngorngo 38007	. . 3 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2		idlval.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
3		idlval.2	. . . 4 ⊢ 𝐻 = (2nd ‘𝑅)
4		idlval.3	. . . 4 ⊢ 𝑋 = ran 𝐺
5		idlval.4	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
6	2, 3, 4, 5	isidl 38021	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
7	1, 6	syl 17	. 2 ⊢ (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
8		ssel2 3978	. . . . . . . 8 ⊢ ((𝐼 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝑋)
9	2, 3, 4	crngocom 38008	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))
10	9	eleq1d 2826	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝑧𝐻𝑥) ∈ 𝐼))
11	10	biimprd 248	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐻𝑥) ∈ 𝐼 → (𝑥𝐻𝑧) ∈ 𝐼))
12	11	3expa 1119	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐻𝑥) ∈ 𝐼 → (𝑥𝐻𝑧) ∈ 𝐼))
13	12	pm4.71d 561	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐻𝑥) ∈ 𝐼 ↔ ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
14	13	bicomd 223	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) ↔ (𝑧𝐻𝑥) ∈ 𝐼))
15	14	ralbidva 3176	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) → (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) ↔ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))
16	15	anbi2d 630	. . . . . . . 8 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) → ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) ↔ (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))
17	8, 16	sylan2 593	. . . . . . 7 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐼 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐼)) → ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) ↔ (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))
18	17	anassrs 467	. . . . . 6 ⊢ (((𝑅 ∈ CRingOps ∧ 𝐼 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐼) → ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) ↔ (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))
19	18	ralbidva 3176	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐼 ⊆ 𝑋) → (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) ↔ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))
20	19	adantrr 717	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼)) → (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) ↔ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))
21	20	pm5.32da 579	. . 3 ⊢ (𝑅 ∈ CRingOps → (((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) ↔ ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))
22		df-3an 1089	. . 3 ⊢ ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) ↔ ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
23		df-3an 1089	. . 3 ⊢ ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)) ↔ ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))
24	21, 22, 23	3bitr4g 314	. 2 ⊢ (𝑅 ∈ CRingOps → ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))
25	7, 24	bitrd 279	1 ⊢ (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ran crn 5686  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  GIdcgi 30509  RingOpscrngo 37901  CRingOpsccring 38000  Idlcidl 38014

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-rngo 37902  df-com2 37997  df-crngo 38001  df-idl 38017

This theorem is referenced by:  prnc  38074