Theorem idllmulcl 37522

Description: An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
idllmulcl.1	⊢ 𝐺 = (1st ‘𝑅)
idllmulcl.2	⊢ 𝐻 = (2nd ‘𝑅)
idllmulcl.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
idllmulcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)

Proof of Theorem idllmulcl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idllmulcl.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		idllmulcl.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
3		idllmulcl.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4		eqid 2725	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 37516	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
6	5	biimpa 475	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
7	6	simp3d 1141	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
8		simpl 481	. . . . . 6 ⊢ (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑧𝐻𝑥) ∈ 𝐼)
9	8	ralimi 3073	. . . . 5 ⊢ (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
10	9	adantl 480	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
11	10	ralimi 3073	. . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
12	7, 11	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
13		oveq2 7422	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑧𝐻𝑥) = (𝑧𝐻𝐴))
14	13	eleq1d 2810	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑧𝐻𝑥) ∈ 𝐼 ↔ (𝑧𝐻𝐴) ∈ 𝐼))
15		oveq1 7421	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧𝐻𝐴) = (𝐵𝐻𝐴))
16	15	eleq1d 2810	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧𝐻𝐴) ∈ 𝐼 ↔ (𝐵𝐻𝐴) ∈ 𝐼))
17	14, 16	rspc2v 3612	. 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼 → (𝐵𝐻𝐴) ∈ 𝐼))
18	12, 17	mpan9 505	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051   ⊆ wss 3939  ran crn 5671  ‘cfv 6541  (class class class)co 7414  1^st c1st 7987  2^nd c2nd 7988  GIdcgi 30316  RingOpscrngo 37396  Idlcidl 37509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6493  df-fun 6543  df-fv 6549  df-ov 7417  df-idl 37512

This theorem is referenced by:  idlnegcl  37524  divrngidl  37530  intidl  37531  unichnidl  37533  prnc  37569  ispridlc  37572