Theorem isidlc 34713

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 34698	. . 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 34712	. . 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 3852	. . . . . . . 8 ⊢ ((𝐼 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝑋)
9	2, 3, 4	crngocom 34699	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))
10	9	eleq1d 2847	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝑧𝐻𝑥) ∈ 𝐼))
11	10	biimprd 240	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐻𝑥) ∈ 𝐼 → (𝑥𝐻𝑧) ∈ 𝐼))
12	11	3expa 1098	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐻𝑥) ∈ 𝐼 → (𝑥𝐻𝑧) ∈ 𝐼))
13	12	pm4.71d 554	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐻𝑥) ∈ 𝐼 ↔ ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
14	13	bicomd 215	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) ↔ (𝑧𝐻𝑥) ∈ 𝐼))
15	14	ralbidva 3143	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) → (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) ↔ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))
16	15	anbi2d 619	. . . . . . . 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 583	. . . . . . 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 460	. . . . . 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 3143	. . . . 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 704	. . . 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 571	. . 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 1070	. . 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 1070	. . 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 306	. 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 271	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 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048  ∀wral 3085   ⊆ wss 3828  ran crn 5405  ‘cfv 6186  (class class class)co 6974  1^st c1st 7496  2^nd c2nd 7497  GIdcgi 28038  RingOpscrngo 34592  CRingOpsccring 34691  Idlcidl 34705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-iota 6150  df-fun 6188  df-fv 6194  df-ov 6977  df-1st 7498  df-2nd 7499  df-rngo 34593  df-com2 34688  df-crngo 34692  df-idl 34708

This theorem is referenced by:  prnc  34765