Theorem rngdi 14214

Description: Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)

Hypotheses

Ref	Expression
rngdi.b	|- B = ( Base ` R )
rngdi.p	|- .+ = ( +g ` R )
rngdi.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
rngdi	|- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )

Proof of Theorem rngdi

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngdi.b	. . . 4 |- B = ( Base ` R )
2		eqid 2238	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
3		rngdi.p	. . . 4 |- .+ = ( +g ` R )
4		rngdi.t	. . . 4 |- .x. = ( .r ` R )
5	1, 2, 3, 4	isrng 14208	. . 3 |- ( R e. Rng <-> ( R e. Abel /\ (mulGrp ` R ) e. Smgrp /\ A. a e. B A. b e. B A. c e. B ( ( a .x. ( b .+ c ) ) = ( ( a .x. b ) .+ ( a .x. c ) ) /\ ( ( a .+ b ) .x. c ) = ( ( a .x. c ) .+ ( b .x. c ) ) ) ) )
6		oveq1 6082	. . . . . . . 8 |- ( a = X -> ( a .x. ( b .+ c ) ) = ( X .x. ( b .+ c ) ) )
7		oveq1 6082	. . . . . . . . 9 |- ( a = X -> ( a .x. b ) = ( X .x. b ) )
8		oveq1 6082	. . . . . . . . 9 |- ( a = X -> ( a .x. c ) = ( X .x. c ) )
9	7, 8	oveq12d 6093	. . . . . . . 8 |- ( a = X -> ( ( a .x. b ) .+ ( a .x. c ) ) = ( ( X .x. b ) .+ ( X .x. c ) ) )
10	6, 9	eqeq12d 2253	. . . . . . 7 |- ( a = X -> ( ( a .x. ( b .+ c ) ) = ( ( a .x. b ) .+ ( a .x. c ) ) <-> ( X .x. ( b .+ c ) ) = ( ( X .x. b ) .+ ( X .x. c ) ) ) )
11		oveq1 6082	. . . . . . . . 9 |- ( a = X -> ( a .+ b ) = ( X .+ b ) )
12	11	oveq1d 6090	. . . . . . . 8 |- ( a = X -> ( ( a .+ b ) .x. c ) = ( ( X .+ b ) .x. c ) )
13	8	oveq1d 6090	. . . . . . . 8 |- ( a = X -> ( ( a .x. c ) .+ ( b .x. c ) ) = ( ( X .x. c ) .+ ( b .x. c ) ) )
14	12, 13	eqeq12d 2253	. . . . . . 7 |- ( a = X -> ( ( ( a .+ b ) .x. c ) = ( ( a .x. c ) .+ ( b .x. c ) ) <-> ( ( X .+ b ) .x. c ) = ( ( X .x. c ) .+ ( b .x. c ) ) ) )
15	10, 14	anbi12d 477	. . . . . 6 |- ( a = X -> ( ( ( a .x. ( b .+ c ) ) = ( ( a .x. b ) .+ ( a .x. c ) ) /\ ( ( a .+ b ) .x. c ) = ( ( a .x. c ) .+ ( b .x. c ) ) ) <-> ( ( X .x. ( b .+ c ) ) = ( ( X .x. b ) .+ ( X .x. c ) ) /\ ( ( X .+ b ) .x. c ) = ( ( X .x. c ) .+ ( b .x. c ) ) ) ) )
16		oveq1 6082	. . . . . . . . 9 |- ( b = Y -> ( b .+ c ) = ( Y .+ c ) )
17	16	oveq2d 6091	. . . . . . . 8 |- ( b = Y -> ( X .x. ( b .+ c ) ) = ( X .x. ( Y .+ c ) ) )
18		oveq2 6083	. . . . . . . . 9 |- ( b = Y -> ( X .x. b ) = ( X .x. Y ) )
19	18	oveq1d 6090	. . . . . . . 8 |- ( b = Y -> ( ( X .x. b ) .+ ( X .x. c ) ) = ( ( X .x. Y ) .+ ( X .x. c ) ) )
20	17, 19	eqeq12d 2253	. . . . . . 7 |- ( b = Y -> ( ( X .x. ( b .+ c ) ) = ( ( X .x. b ) .+ ( X .x. c ) ) <-> ( X .x. ( Y .+ c ) ) = ( ( X .x. Y ) .+ ( X .x. c ) ) ) )
21		oveq2 6083	. . . . . . . . 9 |- ( b = Y -> ( X .+ b ) = ( X .+ Y ) )
22	21	oveq1d 6090	. . . . . . . 8 |- ( b = Y -> ( ( X .+ b ) .x. c ) = ( ( X .+ Y ) .x. c ) )
23		oveq1 6082	. . . . . . . . 9 |- ( b = Y -> ( b .x. c ) = ( Y .x. c ) )
24	23	oveq2d 6091	. . . . . . . 8 |- ( b = Y -> ( ( X .x. c ) .+ ( b .x. c ) ) = ( ( X .x. c ) .+ ( Y .x. c ) ) )
25	22, 24	eqeq12d 2253	. . . . . . 7 |- ( b = Y -> ( ( ( X .+ b ) .x. c ) = ( ( X .x. c ) .+ ( b .x. c ) ) <-> ( ( X .+ Y ) .x. c ) = ( ( X .x. c ) .+ ( Y .x. c ) ) ) )
26	20, 25	anbi12d 477	. . . . . 6 |- ( b = Y -> ( ( ( X .x. ( b .+ c ) ) = ( ( X .x. b ) .+ ( X .x. c ) ) /\ ( ( X .+ b ) .x. c ) = ( ( X .x. c ) .+ ( b .x. c ) ) ) <-> ( ( X .x. ( Y .+ c ) ) = ( ( X .x. Y ) .+ ( X .x. c ) ) /\ ( ( X .+ Y ) .x. c ) = ( ( X .x. c ) .+ ( Y .x. c ) ) ) ) )
27		oveq2 6083	. . . . . . . . 9 |- ( c = Z -> ( Y .+ c ) = ( Y .+ Z ) )
28	27	oveq2d 6091	. . . . . . . 8 |- ( c = Z -> ( X .x. ( Y .+ c ) ) = ( X .x. ( Y .+ Z ) ) )
29		oveq2 6083	. . . . . . . . 9 |- ( c = Z -> ( X .x. c ) = ( X .x. Z ) )
30	29	oveq2d 6091	. . . . . . . 8 |- ( c = Z -> ( ( X .x. Y ) .+ ( X .x. c ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
31	28, 30	eqeq12d 2253	. . . . . . 7 |- ( c = Z -> ( ( X .x. ( Y .+ c ) ) = ( ( X .x. Y ) .+ ( X .x. c ) ) <-> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) )
32		oveq2 6083	. . . . . . . 8 |- ( c = Z -> ( ( X .+ Y ) .x. c ) = ( ( X .+ Y ) .x. Z ) )
33		oveq2 6083	. . . . . . . . 9 |- ( c = Z -> ( Y .x. c ) = ( Y .x. Z ) )
34	29, 33	oveq12d 6093	. . . . . . . 8 |- ( c = Z -> ( ( X .x. c ) .+ ( Y .x. c ) ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) )
35	32, 34	eqeq12d 2253	. . . . . . 7 |- ( c = Z -> ( ( ( X .+ Y ) .x. c ) = ( ( X .x. c ) .+ ( Y .x. c ) ) <-> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) )
36	31, 35	anbi12d 477	. . . . . 6 |- ( c = Z -> ( ( ( X .x. ( Y .+ c ) ) = ( ( X .x. Y ) .+ ( X .x. c ) ) /\ ( ( X .+ Y ) .x. c ) = ( ( X .x. c ) .+ ( Y .x. c ) ) ) <-> ( ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) /\ ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) ) )
37	15, 26, 36	rspc3v 2946	. . . . 5 |- ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( A. a e. B A. b e. B A. c e. B ( ( a .x. ( b .+ c ) ) = ( ( a .x. b ) .+ ( a .x. c ) ) /\ ( ( a .+ b ) .x. c ) = ( ( a .x. c ) .+ ( b .x. c ) ) ) -> ( ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) /\ ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) ) )
38		simpl 109	. . . . 5 |- ( ( ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) /\ ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
39	37, 38	syl6com 35	. . . 4 |- ( A. a e. B A. b e. B A. c e. B ( ( a .x. ( b .+ c ) ) = ( ( a .x. b ) .+ ( a .x. c ) ) /\ ( ( a .+ b ) .x. c ) = ( ( a .x. c ) .+ ( b .x. c ) ) ) -> ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) )
40	39	3ad2ant3 1051	. . 3 |- ( ( R e. Abel /\ (mulGrp ` R ) e. Smgrp /\ A. a e. B A. b e. B A. c e. B ( ( a .x. ( b .+ c ) ) = ( ( a .x. b ) .+ ( a .x. c ) ) /\ ( ( a .+ b ) .x. c ) = ( ( a .x. c ) .+ ( b .x. c ) ) ) ) -> ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) )
41	5, 40	sylbi 121	. 2 |- ( R e. Rng -> ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) )
42	41	imp 124	1 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409  Smgrpcsgrp 13693   Abelcabl 14065  mulGrpcmgp 14194  Rngcrng 14206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mulr 13422  df-rng 14207

This theorem is referenced by:  rngrz  14220  rngmneg2  14222  rngsubdi  14225  rngressid  14228  imasrng  14230  opprrng  14355  issubrng2  14491  rnglidlrng  14807