Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngocl Structured version   Visualization version   GIF version

Theorem rngocl 34030
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 34029 . 2 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5 fovrn 7044 . 2 ((𝐻:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
64, 5syl3an1 1195 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100   = wceq 1637  wcel 2157   × cxp 5322  ran crn 5325  wf 6107  cfv 6111  (class class class)co 6884  1st c1st 7406  2nd c2nd 7407  RingOpscrngo 34023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-fv 6119  df-ov 6887  df-1st 7408  df-2nd 7409  df-rngo 34024
This theorem is referenced by:  rngolz  34051  rngorz  34052  rngonegmn1l  34070  rngonegmn1r  34071  rngoneglmul  34072  rngonegrmul  34073  rngosubdi  34074  rngosubdir  34075  isdrngo2  34087  rngohomco  34103  rngoisocnv  34110  crngm4  34132  rngoidl  34153  keridl  34161  prnc  34196  ispridlc  34199  pridlc3  34202  dmncan1  34205
  Copyright terms: Public domain W3C validator