Theorem idllmulcl 38260

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 2737	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 2, 3, 4	isidl 38254	. . . . 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 1145	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
8		simpl 482	. . . . . 6 ⊢ (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑧𝐻𝑥) ∈ 𝐼)
9	8	ralimi 3075	. . . . 5 ⊢ (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
10	9	adantl 481	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
11	10	ralimi 3075	. . 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 7376	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑧𝐻𝑥) = (𝑧𝐻𝐴))
14	13	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑧𝐻𝑥) ∈ 𝐼 ↔ (𝑧𝐻𝐴) ∈ 𝐼))
15		oveq1 7375	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧𝐻𝐴) = (𝐵𝐻𝐴))
16	15	eleq1d 2822	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧𝐻𝐴) ∈ 𝐼 ↔ (𝐵𝐻𝐴) ∈ 𝐼))
17	14, 16	rspc2v 3589	. 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼 → (𝐵𝐻𝐴) ∈ 𝐼))
18	12, 17	mpan9 506	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  ran crn 5633  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  GIdcgi 30577  RingOpscrngo 38134  Idlcidl 38247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-idl 38250

This theorem is referenced by:  idlnegcl  38262  divrngidl  38268  intidl  38269  unichnidl  38271  prnc  38307  ispridlc  38310