Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idlrmulcl Structured version   Visualization version   GIF version

Theorem idlrmulcl 37402
Description: An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idllmulcl.1 𝐺 = (1st𝑅)
idllmulcl.2 𝐻 = (2nd𝑅)
idllmulcl.3 𝑋 = ran 𝐺
Assertion
Ref Expression
idlrmulcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)

Proof of Theorem idlrmulcl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idllmulcl.1 . . . . . 6 𝐺 = (1st𝑅)
2 idllmulcl.2 . . . . . 6 𝐻 = (2nd𝑅)
3 idllmulcl.3 . . . . . 6 𝑋 = ran 𝐺
4 eqid 2726 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 37395 . . . . 5 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
65biimpa 476 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
76simp3d 1141 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
8 simpr 484 . . . . . 6 (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑥𝐻𝑧) ∈ 𝐼)
98ralimi 3077 . . . . 5 (∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
109adantl 481 . . . 4 ((∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
1110ralimi 3077 . . 3 (∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑥𝐼𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
127, 11syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥𝐼𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
13 oveq1 7412 . . . 4 (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
1413eleq1d 2812 . . 3 (𝑥 = 𝐴 → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝑧) ∈ 𝐼))
15 oveq2 7413 . . . 4 (𝑧 = 𝐵 → (𝐴𝐻𝑧) = (𝐴𝐻𝐵))
1615eleq1d 2812 . . 3 (𝑧 = 𝐵 → ((𝐴𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝐵) ∈ 𝐼))
1714, 16rspc2v 3617 . 2 ((𝐴𝐼𝐵𝑋) → (∀𝑥𝐼𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼 → (𝐴𝐻𝐵) ∈ 𝐼))
1812, 17mpan9 506 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wss 3943  ran crn 5670  cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  GIdcgi 30252  RingOpscrngo 37275  Idlcidl 37388
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-idl 37391
This theorem is referenced by:  1idl  37407  intidl  37410  unichnidl  37412
  Copyright terms: Public domain W3C validator