Theorem idlrmulcl 37981

Description: An ideal is closed under multiplication on the right. (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
idlrmulcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)

Proof of Theorem idlrmulcl

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 2740	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 37974	. . . . 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 476	. . . 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 1144	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
8		simpr 484	. . . . . 6 ⊢ (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑥𝐻𝑧) ∈ 𝐼)
9	8	ralimi 3089	. . . . 5 ⊢ (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
10	9	adantl 481	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
11	10	ralimi 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
12	7, 11	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
13		oveq1 7455	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
14	13	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝑧) ∈ 𝐼))
15		oveq2 7456	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴𝐻𝑧) = (𝐴𝐻𝐵))
16	15	eleq1d 2829	. . 3 ⊢ (𝑧 = 𝐵 → ((𝐴𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝐵) ∈ 𝐼))
17	14, 16	rspc2v 3646	. 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼 → (𝐴𝐻𝐵) ∈ 𝐼))
18	12, 17	mpan9 506	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)

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:  1idl  37986  intidl  37989  unichnidl  37991