Theorem rngocl 38552

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 38551	. 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5		fovcdm 7580	. 2 ⊢ ((𝐻:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
6	4, 5	syl3an1 1181	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   × cxp 5659  ran crn 5662  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  RingOpscrngo 38545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-1st 7982  df-2nd 7983  df-rngo 38546

This theorem is referenced by:  rngolz  38573  rngorz  38574  rngonegmn1l  38592  rngonegmn1r  38593  rngoneglmul  38594  rngonegrmul  38595  rngosubdi  38596  rngosubdir  38597  isdrngo2  38609  rngohomco  38625  rngoisocnv  38632  crngm4  38654  rngoidl  38675  keridl  38683  prnc  38718  ispridlc  38721  pridlc3  38724  dmncan1  38727