Theorem idladdcl 36104

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 2738	. . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2738	. . . . . 6 ⊢ ran 𝐺 = ran 𝐺
4		eqid 2738	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 36099	. . . . 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 1142	. . 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 3086	. . 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 7262	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
12	11	eleq1d 2823	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝑦) ∈ 𝐼))
13		oveq2 7263	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
14	13	eleq1d 2823	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝐵) ∈ 𝐼))
15	12, 14	rspc2v 3562	. 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 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  ran crn 5581  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  GIdcgi 28753  RingOpscrngo 35979  Idlcidl 36092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-idl 36095

This theorem is referenced by:  idlsubcl  36108  intidl  36114  unichnidl  36116