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Theorem idlrmulcl 35452
 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 2801 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 35445 . . . . 5 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
65biimpa 480 . . . 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 488 . . . . . 6 (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑥𝐻𝑧) ∈ 𝐼)
98ralimi 3131 . . . . 5 (∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
109adantl 485 . . . 4 ((∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼)
1110ralimi 3131 . . 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 7146 . . . 4 (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
1413eleq1d 2877 . . 3 (𝑥 = 𝐴 → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝑧) ∈ 𝐼))
15 oveq2 7147 . . . 4 (𝑧 = 𝐵 → (𝐴𝐻𝑧) = (𝐴𝐻𝐵))
1615eleq1d 2877 . . 3 (𝑧 = 𝐵 → ((𝐴𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝐵) ∈ 𝐼))
1714, 16rspc2v 3584 . 2 ((𝐴𝐼𝐵𝑋) → (∀𝑥𝐼𝑧𝑋 (𝑥𝐻𝑧) ∈ 𝐼 → (𝐴𝐻𝐵) ∈ 𝐼))
1812, 17mpan9 510 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  ran crn 5524  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  GIdcgi 28276  RingOpscrngo 35325  Idlcidl 35438 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-idl 35441 This theorem is referenced by:  1idl  35457  intidl  35460  unichnidl  35462
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