Theorem idladdcl 37525

Description: An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypothesis

Ref	Expression
idladdcl.1	⊢ 𝐺 = (1st ‘𝑅)

Assertion

Ref	Expression
idladdcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)

Proof of Theorem idladdcl

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

Step	Hyp	Ref	Expression
1		idladdcl.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2728	. . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2728	. . . . . 6 ⊢ ran 𝐺 = ran 𝐺
4		eqid 2728	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 37520	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ ran 𝐺 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)))))
6	5	biimpa 475	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ ran 𝐺 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼))))
7	6	simp3d 1141	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)))
8		simpl 481	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)) → ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼)
9	8	ralimi 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)) → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼)
10	7, 9	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼)
11		oveq1 7433	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
12	11	eleq1d 2814	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝑦) ∈ 𝐼))
13		oveq2 7434	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
14	13	eleq1d 2814	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝐵) ∈ 𝐼))
15	12, 14	rspc2v 3622	. 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼) → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 → (𝐴𝐺𝐵) ∈ 𝐼))
16	10, 15	mpan9 505	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058   ⊆ wss 3949  ran crn 5683  ‘cfv 6553  (class class class)co 7426  1^st c1st 7997  2^nd c2nd 7998  GIdcgi 30320  RingOpscrngo 37400  Idlcidl 37513

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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-idl 37516

This theorem is referenced by:  idlsubcl  37529  intidl  37535  unichnidl  37537