Theorem rngsubdi 13788

Description: Ring multiplication distributes over subtraction. (subdi 8477 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 13893. (Revised by AV, 23-Feb-2025.)

Hypotheses

Ref	Expression
rngsubdi.b	|- B = ( Base ` R )
rngsubdi.t	|- .x. = ( .r ` R )
rngsubdi.m	|- .- = ( -g ` R )
rngsubdi.r	|- ( ph -> R e. Rng )
rngsubdi.x	|- ( ph -> X e. B )
rngsubdi.y	|- ( ph -> Y e. B )
rngsubdi.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
rngsubdi	|- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )

Proof of Theorem rngsubdi

Step	Hyp	Ref	Expression
1		rngsubdi.r	. . . 4 |- ( ph -> R e. Rng )
2		rngsubdi.x	. . . 4 |- ( ph -> X e. B )
3		rngsubdi.y	. . . 4 |- ( ph -> Y e. B )
4		rngsubdi.b	. . . . 5 |- B = ( Base ` R )
5		eqid 2206	. . . . 5 |- ( invg ` R ) = ( invg ` R )
6		rnggrp 13775	. . . . . 6 |- ( R e. Rng -> R e. Grp )
7	1, 6	syl 14	. . . . 5 |- ( ph -> R e. Grp )
8		rngsubdi.z	. . . . 5 |- ( ph -> Z e. B )
9	4, 5, 7, 8	grpinvcld 13456	. . . 4 |- ( ph -> ( ( invg ` R ) ` Z ) e. B )
10		eqid 2206	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		rngsubdi.t	. . . . 5 |- .x. = ( .r ` R )
12	4, 10, 11	rngdi 13777	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ ( ( invg ` R ) ` Z ) e. B ) ) -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) )
13	1, 2, 3, 9, 12	syl13anc 1252	. . 3 |- ( ph -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) )
14	4, 11, 5, 1, 2, 8	rngmneg2 13785	. . . 4 |- ( ph -> ( X .x. ( ( invg ` R ) ` Z ) ) = ( ( invg ` R ) ` ( X .x. Z ) ) )
15	14	oveq2d 5973	. . 3 |- ( ph -> ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
16	13, 15	eqtrd 2239	. 2 |- ( ph -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
17		rngsubdi.m	. . . . 5 |- .- = ( -g ` R )
18	4, 10, 5, 17	grpsubval 13453	. . . 4 |- ( ( Y e. B /\ Z e. B ) -> ( Y .- Z ) = ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) )
19	3, 8, 18	syl2anc 411	. . 3 |- ( ph -> ( Y .- Z ) = ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) )
20	19	oveq2d 5973	. 2 |- ( ph -> ( X .x. ( Y .- Z ) ) = ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) )
21	4, 11	rngcl 13781	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
22	1, 2, 3, 21	syl3anc 1250	. . 3 |- ( ph -> ( X .x. Y ) e. B )
23	4, 11	rngcl 13781	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
24	1, 2, 8, 23	syl3anc 1250	. . 3 |- ( ph -> ( X .x. Z ) e. B )
25	4, 10, 5, 17	grpsubval 13453	. . 3 |- ( ( ( X .x. Y ) e. B /\ ( X .x. Z ) e. B ) -> ( ( X .x. Y ) .- ( X .x. Z ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
26	22, 24, 25	syl2anc 411	. 2 |- ( ph -> ( ( X .x. Y ) .- ( X .x. Z ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
27	16, 20, 26	3eqtr4d 2249	1 |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   .rcmulr 12985   Grpcgrp 13407   invgcminusg 13408   -gcsg 13409  Rngcrng 13769

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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-plusg 12997  df-mulr 12998  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-sbg 13412  df-abl 13698  df-mgp 13758  df-rng 13770

This theorem is referenced by:  2idlcpblrng  14360