Theorem idladdcl 37971

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 2740	. . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2740	. . . . . 6 ⊢ ran 𝐺 = ran 𝐺
4		eqid 2740	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 37966	. . . . 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 476	. . . 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 1144	. . 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 482	. . . 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 3089	. . 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 7450	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
12	11	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝑦) ∈ 𝐼))
13		oveq2 7451	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
14	13	eleq1d 2829	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝐵) ∈ 𝐼))
15	12, 14	rspc2v 3646	. 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼) → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 → (𝐴𝐺𝐵) ∈ 𝐼))
16	10, 15	mpan9 506	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ran crn 5696  ‘cfv 6568  (class class class)co 7443  1^st c1st 8022  2^nd c2nd 8023  GIdcgi 30514  RingOpscrngo 37846  Idlcidl 37959

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 7764

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 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-iota 6520  df-fun 6570  df-fv 6576  df-ov 7446  df-idl 37962

This theorem is referenced by:  idlsubcl  37975  intidl  37981  unichnidl  37983