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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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