Theorem mulgsubdir 13067

Description: Distribution of group multiples over subtraction for group elements, subdir 8360 analog. (Contributed by Mario Carneiro, 13-Dec-2014.)

Hypotheses

Ref	Expression
mulgsubdir.b	|- B = ( Base ` G )
mulgsubdir.t	|- .x. = (.g ` G )
mulgsubdir.d	|- .- = ( -g ` G )

Assertion

Ref	Expression
mulgsubdir	|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M - N ) .x. X ) = ( ( M .x. X ) .- ( N .x. X ) ) )

Proof of Theorem mulgsubdir

Step	Hyp	Ref	Expression
1		znegcl 9301	. . 3 |- ( N e. ZZ -> -u N e. ZZ )
2		mulgsubdir.b	. . . 4 |- B = ( Base ` G )
3		mulgsubdir.t	. . . 4 |- .x. = (.g ` G )
4		eqid 2188	. . . 4 |- ( +g ` G ) = ( +g ` G )
5	2, 3, 4	mulgdir 13059	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ -u N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) )
6	1, 5	syl3anr2 1301	. 2 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) )
7		simpr1 1004	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. ZZ )
8	7	zcnd 9393	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. CC )
9		simpr2 1005	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. ZZ )
10	9	zcnd 9393	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. CC )
11	8, 10	negsubd 8291	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M + -u N ) = ( M - N ) )
12	11	oveq1d 5905	. 2 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M - N ) .x. X ) )
13		eqid 2188	. . . . . 6 |- ( invg ` G ) = ( invg ` G )
14	2, 3, 13	mulgneg 13045	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
15	14	3adant3r1 1213	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
16	15	oveq2d 5906	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) = ( ( M .x. X ) ( +g ` G ) ( ( invg ` G ) ` ( N .x. X ) ) ) )
17	2, 3	mulgcl 13044	. . . . 5 |- ( ( G e. Grp /\ M e. ZZ /\ X e. B ) -> ( M .x. X ) e. B )
18	17	3adant3r2 1214	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M .x. X ) e. B )
19	2, 3	mulgcl 13044	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
20	19	3adant3r1 1213	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( N .x. X ) e. B )
21		mulgsubdir.d	. . . . 5 |- .- = ( -g ` G )
22	2, 4, 13, 21	grpsubval 12955	. . . 4 |- ( ( ( M .x. X ) e. B /\ ( N .x. X ) e. B ) -> ( ( M .x. X ) .- ( N .x. X ) ) = ( ( M .x. X ) ( +g ` G ) ( ( invg ` G ) ` ( N .x. X ) ) ) )
23	18, 20, 22	syl2anc 411	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M .x. X ) .- ( N .x. X ) ) = ( ( M .x. X ) ( +g ` G ) ( ( invg ` G ) ` ( N .x. X ) ) ) )
24	16, 23	eqtr4d 2224	. 2 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) = ( ( M .x. X ) .- ( N .x. X ) ) )
25	6, 12, 24	3eqtr3d 2229	1 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M - N ) .x. X ) = ( ( M .x. X ) .- ( N .x. X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 979   = wceq 1363   e. wcel 2159   ` cfv 5230  (class class class)co 5890   + caddc 7831   - cmin 8145   -ucneg 8146   ZZcz 9270   Basecbs 12479   +g cplusg 12554   Grpcgrp 12910   invgcminusg 12911   -gcsg 12912  ._gcmg 13026

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-0id 7936  ax-rnegex 7937  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-ltadd 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-inn 8937  df-2 8995  df-n0 9194  df-z 9271  df-uz 9546  df-fz 10026  df-seqfrec 10463  df-ndx 12482  df-slot 12483  df-base 12485  df-plusg 12567  df-0g 12728  df-mgm 12797  df-sgrp 12830  df-mnd 12843  df-grp 12913  df-minusg 12914  df-sbg 12915  df-mulg 13027

This theorem is referenced by: (None)