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Theorem rngocl 37072
Description: Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1 𝐺 = (1st𝑅)
ringi.2 𝐻 = (2nd𝑅)
ringi.3 𝑋 = ran 𝐺
Assertion
Ref Expression
rngocl ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)

Proof of Theorem rngocl
StepHypRef Expression
1 ringi.1 . . 3 𝐺 = (1st𝑅)
2 ringi.2 . . 3 𝐻 = (2nd𝑅)
3 ringi.3 . . 3 𝑋 = ran 𝐺
41, 2, 3rngosm 37071 . 2 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5 fovcdm 7579 . 2 ((𝐻:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
64, 5syl3an1 1161 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1539  wcel 2104   × cxp 5673  ran crn 5676  wf 6538  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  RingOpscrngo 37065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-1st 7977  df-2nd 7978  df-rngo 37066
This theorem is referenced by:  rngolz  37093  rngorz  37094  rngonegmn1l  37112  rngonegmn1r  37113  rngoneglmul  37114  rngonegrmul  37115  rngosubdi  37116  rngosubdir  37117  isdrngo2  37129  rngohomco  37145  rngoisocnv  37152  crngm4  37174  rngoidl  37195  keridl  37203  prnc  37238  ispridlc  37241  pridlc3  37244  dmncan1  37247
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