Theorem rngocl 38181

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 38180	. 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
5		fovcdm 7540	. 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 5632  ran crn 5635  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370  1^st c1st 7943  2^nd c2nd 7944  RingOpscrngo 38174

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 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-ov 7373  df-1st 7945  df-2nd 7946  df-rngo 38175

This theorem is referenced by:  rngolz  38202  rngorz  38203  rngonegmn1l  38221  rngonegmn1r  38222  rngoneglmul  38223  rngonegrmul  38224  rngosubdi  38225  rngosubdir  38226  isdrngo2  38238  rngohomco  38254  rngoisocnv  38261  crngm4  38283  rngoidl  38304  keridl  38312  prnc  38347  ispridlc  38350  pridlc3  38353  dmncan1  38356