Theorem mulgsubdir 13694

Description: Distribution of group multiples over subtraction for group elements, subdir 8528 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 9473	. . 3 |- ( N e. ZZ -> -u N e. ZZ )
2		mulgsubdir.b	. . . 4 |- B = ( Base ` G )
3		mulgsubdir.t	. . . 4 |- .x. = (.g ` G )
4		eqid 2229	. . . 4 |- ( +g ` G ) = ( +g ` G )
5	2, 3, 4	mulgdir 13686	. . 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 1324	. 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 1027	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. ZZ )
8	7	zcnd 9566	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. CC )
9		simpr2 1028	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. ZZ )
10	9	zcnd 9566	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. CC )
11	8, 10	negsubd 8459	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M + -u N ) = ( M - N ) )
12	11	oveq1d 6015	. 2 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M - N ) .x. X ) )
13		eqid 2229	. . . . . 6 |- ( invg ` G ) = ( invg ` G )
14	2, 3, 13	mulgneg 13672	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
15	14	3adant3r1 1236	. . . 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 6016	. . 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 13671	. . . . 5 |- ( ( G e. Grp /\ M e. ZZ /\ X e. B ) -> ( M .x. X ) e. B )
18	17	3adant3r2 1237	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M .x. X ) e. B )
19	2, 3	mulgcl 13671	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
20	19	3adant3r1 1236	. . . 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 13574	. . . 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 2265	. 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 2270	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 1002   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   + caddc 7998   - cmin 8313   -ucneg 8314   ZZcz 9442   Basecbs 13027   +g cplusg 13105   Grpcgrp 13528   invgcminusg 13529   -gcsg 13530  ._gcmg 13651

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-mulg 13652

This theorem is referenced by: (None)