Theorem rngocl 33700

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 e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. 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 33699	. 2 |- ( R e. RingOps -> H : ( X X. X ) --> X )
5		fovrn 6804	. 2 |- ( ( H : ( X X. X ) --> X /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
6	4, 5	syl3an1 1359	1 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694

This theorem is referenced by:  rngolz  33721  rngorz  33722  rngonegmn1l  33740  rngonegmn1r  33741  rngoneglmul  33742  rngonegrmul  33743  rngosubdi  33744  rngosubdir  33745  isdrngo2  33757  rngohomco  33773  rngoisocnv  33780  crngm4  33802  rngoidl  33823  keridl  33831  prnc  33866  ispridlc  33869  pridlc3  33872  dmncan1  33875