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Theorem idllmulcl 35336
Description: An ideal is closed under multiplication on the left. (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
idllmulcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)

Proof of Theorem idllmulcl
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 2821 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 35330 . . . . 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 simpl 486 . . . . . 6 (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑧𝐻𝑥) ∈ 𝐼)
98ralimi 3148 . . . . 5 (∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
109adantl 485 . . . 4 ((∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
1110ralimi 3148 . . 3 (∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑥𝐼𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
127, 11syl 17 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥𝐼𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
13 oveq2 7138 . . . 4 (𝑥 = 𝐴 → (𝑧𝐻𝑥) = (𝑧𝐻𝐴))
1413eleq1d 2896 . . 3 (𝑥 = 𝐴 → ((𝑧𝐻𝑥) ∈ 𝐼 ↔ (𝑧𝐻𝐴) ∈ 𝐼))
15 oveq1 7137 . . . 4 (𝑧 = 𝐵 → (𝑧𝐻𝐴) = (𝐵𝐻𝐴))
1615eleq1d 2896 . . 3 (𝑧 = 𝐵 → ((𝑧𝐻𝐴) ∈ 𝐼 ↔ (𝐵𝐻𝐴) ∈ 𝐼))
1714, 16rspc2v 3610 . 2 ((𝐴𝐼𝐵𝑋) → (∀𝑥𝐼𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼 → (𝐵𝐻𝐴) ∈ 𝐼))
1812, 17mpan9 510 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  wss 3910  ran crn 5529  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  GIdcgi 28251  RingOpscrngo 35210  Idlcidl 35323
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-idl 35326
This theorem is referenced by:  idlnegcl  35338  divrngidl  35344  intidl  35345  unichnidl  35347  prnc  35383  ispridlc  35386
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