Theorem rngocl 37908

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 = (1st ‘𝑅)
ringi.2	⊢ 𝐻 = (2nd ‘𝑅)
ringi.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngocl	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)

Proof of Theorem rngocl

Step	Hyp	Ref	Expression
1		ringi.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
3		ringi.3	. . 3 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	rngosm 37907	. 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5		fovcdm 7603	. 2 ⊢ ((𝐻:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
6	4, 5	syl3an1 1164	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   × cxp 5683  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  RingOpscrngo 37901

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-rngo 37902

This theorem is referenced by:  rngolz  37929  rngorz  37930  rngonegmn1l  37948  rngonegmn1r  37949  rngoneglmul  37950  rngonegrmul  37951  rngosubdi  37952  rngosubdir  37953  isdrngo2  37965  rngohomco  37981  rngoisocnv  37988  crngm4  38010  rngoidl  38031  keridl  38039  prnc  38074  ispridlc  38077  pridlc3  38080  dmncan1  38083