Theorem rngodi 33703

Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (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
rngodi	|- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B G C ) ) = ( ( A H B ) G ( A H C ) ) )

Proof of Theorem rngodi

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringi.1	. . . . 5 |- G = ( 1st ` R )
2		ringi.2	. . . . 5 |- H = ( 2nd ` R )
3		ringi.3	. . . . 5 |- X = ran G
4	1, 2, 3	rngoi 33698	. . . 4 |- ( R e. RingOps -> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) ) )
5	4	simprd 479	. . 3 |- ( R e. RingOps -> ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) )
6	5	simpld 475	. 2 |- ( R e. RingOps -> A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) )
7		simp2 1062	. . . . 5 |- ( ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) )
8	7	ralimi 2952	. . . 4 |- ( A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> A. z e. X ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) )
9	8	2ralimi 2953	. . 3 |- ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> A. x e. X A. y e. X A. z e. X ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) )
10		oveq1 6657	. . . . 5 |- ( x = A -> ( x H ( y G z ) ) = ( A H ( y G z ) ) )
11		oveq1 6657	. . . . . 6 |- ( x = A -> ( x H y ) = ( A H y ) )
12		oveq1 6657	. . . . . 6 |- ( x = A -> ( x H z ) = ( A H z ) )
13	11, 12	oveq12d 6668	. . . . 5 |- ( x = A -> ( ( x H y ) G ( x H z ) ) = ( ( A H y ) G ( A H z ) ) )
14	10, 13	eqeq12d 2637	. . . 4 |- ( x = A -> ( ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) <-> ( A H ( y G z ) ) = ( ( A H y ) G ( A H z ) ) ) )
15		oveq1 6657	. . . . . 6 |- ( y = B -> ( y G z ) = ( B G z ) )
16	15	oveq2d 6666	. . . . 5 |- ( y = B -> ( A H ( y G z ) ) = ( A H ( B G z ) ) )
17		oveq2 6658	. . . . . 6 |- ( y = B -> ( A H y ) = ( A H B ) )
18	17	oveq1d 6665	. . . . 5 |- ( y = B -> ( ( A H y ) G ( A H z ) ) = ( ( A H B ) G ( A H z ) ) )
19	16, 18	eqeq12d 2637	. . . 4 |- ( y = B -> ( ( A H ( y G z ) ) = ( ( A H y ) G ( A H z ) ) <-> ( A H ( B G z ) ) = ( ( A H B ) G ( A H z ) ) ) )
20		oveq2 6658	. . . . . 6 |- ( z = C -> ( B G z ) = ( B G C ) )
21	20	oveq2d 6666	. . . . 5 |- ( z = C -> ( A H ( B G z ) ) = ( A H ( B G C ) ) )
22		oveq2 6658	. . . . . 6 |- ( z = C -> ( A H z ) = ( A H C ) )
23	22	oveq2d 6666	. . . . 5 |- ( z = C -> ( ( A H B ) G ( A H z ) ) = ( ( A H B ) G ( A H C ) ) )
24	21, 23	eqeq12d 2637	. . . 4 |- ( z = C -> ( ( A H ( B G z ) ) = ( ( A H B ) G ( A H z ) ) <-> ( A H ( B G C ) ) = ( ( A H B ) G ( A H C ) ) ) )
25	14, 19, 24	rspc3v 3325	. . 3 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( A. x e. X A. y e. X A. z e. X ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) -> ( A H ( B G C ) ) = ( ( A H B ) G ( A H C ) ) ) )
26	9, 25	syl5 34	. 2 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) -> ( A H ( B G C ) ) = ( ( A H B ) G ( A H C ) ) ) )
27	6, 26	mpan9 486	1 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B G C ) ) = ( ( A H B ) G ( A H C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   AbelOpcablo 27398   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:  rngorz  33722  rngonegmn1r  33741  rngosubdi  33744