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Theorem idllmulcl 34305
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 2800 . . . . . 6 (GId‘𝐺) = (GId‘𝐺)
51, 2, 3, 4isidl 34299 . . . . 5 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
65biimpa 469 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
76simp3d 1175 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
8 simpl 475 . . . . . 6 (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑧𝐻𝑥) ∈ 𝐼)
98ralimi 3134 . . . . 5 (∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
109adantl 474 . . . 4 ((∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼)
1110ralimi 3134 . . 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 6887 . . . 4 (𝑥 = 𝐴 → (𝑧𝐻𝑥) = (𝑧𝐻𝐴))
1413eleq1d 2864 . . 3 (𝑥 = 𝐴 → ((𝑧𝐻𝑥) ∈ 𝐼 ↔ (𝑧𝐻𝐴) ∈ 𝐼))
15 oveq1 6886 . . . 4 (𝑧 = 𝐵 → (𝑧𝐻𝐴) = (𝐵𝐻𝐴))
1615eleq1d 2864 . . 3 (𝑧 = 𝐵 → ((𝑧𝐻𝐴) ∈ 𝐼 ↔ (𝐵𝐻𝐴) ∈ 𝐼))
1714, 16rspc2v 3511 . 2 ((𝐴𝐼𝐵𝑋) → (∀𝑥𝐼𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼 → (𝐵𝐻𝐴) ∈ 𝐼))
1812, 17mpan9 503 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3090  wss 3770  ran crn 5314  cfv 6102  (class class class)co 6879  1st c1st 7400  2nd c2nd 7401  GIdcgi 27869  RingOpscrngo 34179  Idlcidl 34292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-idl 34295
This theorem is referenced by:  idlnegcl  34307  divrngidl  34313  intidl  34314  unichnidl  34316  prnc  34352  ispridlc  34355
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