Theorem rngocl 38239

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 38238	. 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5		fovcdm 7531	. 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 1542   ∈ wcel 2114   × cxp 5623  ran crn 5626  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  RingOpscrngo 38232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-1st 7936  df-2nd 7937  df-rngo 38233

This theorem is referenced by:  rngolz  38260  rngorz  38261  rngonegmn1l  38279  rngonegmn1r  38280  rngoneglmul  38281  rngonegrmul  38282  rngosubdi  38283  rngosubdir  38284  isdrngo2  38296  rngohomco  38312  rngoisocnv  38319  crngm4  38341  rngoidl  38362  keridl  38370  prnc  38405  ispridlc  38408  pridlc3  38411  dmncan1  38414