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Theorem rngocl 34031
 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 34030 . 2 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5 fovrn 6970 . 2 ((𝐻:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
64, 5syl3an1 1167 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   × cxp 5264  ran crn 5267  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  RingOpscrngo 34024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-1st 7334  df-2nd 7335  df-rngo 34025 This theorem is referenced by:  rngolz  34052  rngorz  34053  rngonegmn1l  34071  rngonegmn1r  34072  rngoneglmul  34073  rngonegrmul  34074  rngosubdi  34075  rngosubdir  34076  isdrngo2  34088  rngohomco  34104  rngoisocnv  34111  crngm4  34133  rngoidl  34154  keridl  34162  prnc  34197  ispridlc  34200  pridlc3  34203  dmncan1  34206
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