Theorem mulgsubdir 13023

Description: Distribution of group multiples over subtraction for group elements, subdir 8343 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 9284	. . 3 |- ( N e. ZZ -> -u N e. ZZ )
2		mulgsubdir.b	. . . 4 |- B = ( Base ` G )
3		mulgsubdir.t	. . . 4 |- .x. = (.g ` G )
4		eqid 2177	. . . 4 |- ( +g ` G ) = ( +g ` G )
5	2, 3, 4	mulgdir 13015	. . 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 1291	. 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 1003	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. ZZ )
8	7	zcnd 9376	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. CC )
9		simpr2 1004	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. ZZ )
10	9	zcnd 9376	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. CC )
11	8, 10	negsubd 8274	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M + -u N ) = ( M - N ) )
12	11	oveq1d 5890	. 2 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M - N ) .x. X ) )
13		eqid 2177	. . . . . 6 |- ( invg ` G ) = ( invg ` G )
14	2, 3, 13	mulgneg 13001	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
15	14	3adant3r1 1212	. . . 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 5891	. . 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 13000	. . . . 5 |- ( ( G e. Grp /\ M e. ZZ /\ X e. B ) -> ( M .x. X ) e. B )
18	17	3adant3r2 1213	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M .x. X ) e. B )
19	2, 3	mulgcl 13000	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
20	19	3adant3r1 1212	. . . 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 12919	. . . 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 2213	. 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 2218	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 978   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   + caddc 7814   - cmin 8128   -ucneg 8129   ZZcz 9253   Basecbs 12462   +g cplusg 12536   Grpcgrp 12877   invgcminusg 12878   -gcsg 12879  ._gcmg 12983

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-2 8978  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009  df-seqfrec 10446  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-sbg 12882  df-mulg 12984

This theorem is referenced by: (None)