Theorem mulgsubdir 12878

Description: Distribution of group multiples over subtraction for group elements, subdir 8314 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 9252	. . 3 |- ( N e. ZZ -> -u N e. ZZ )
2		mulgsubdir.b	. . . 4 |- B = ( Base ` G )
3		mulgsubdir.t	. . . 4 |- .x. = (.g ` G )
4		eqid 2173	. . . 4 |- ( +g ` G ) = ( +g ` G )
5	2, 3, 4	mulgdir 12870	. . 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 1289	. 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 1001	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. ZZ )
8	7	zcnd 9344	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. CC )
9		simpr2 1002	. . . . 5 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. ZZ )
10	9	zcnd 9344	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. CC )
11	8, 10	negsubd 8245	. . 3 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M + -u N ) = ( M - N ) )
12	11	oveq1d 5877	. 2 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M - N ) .x. X ) )
13		eqid 2173	. . . . . 6 |- ( invg ` G ) = ( invg ` G )
14	2, 3, 13	mulgneg 12857	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
15	14	3adant3r1 1210	. . . 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 5878	. . 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 12856	. . . . 5 |- ( ( G e. Grp /\ M e. ZZ /\ X e. B ) -> ( M .x. X ) e. B )
18	17	3adant3r2 1211	. . . 4 |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M .x. X ) e. B )
19	2, 3	mulgcl 12856	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
20	19	3adant3r1 1210	. . . 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 12776	. . . 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 2209	. 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 2214	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 976   = wceq 1351   e. wcel 2144   ` cfv 5205  (class class class)co 5862   + caddc 7786   - cmin 8099   -ucneg 8100   ZZcz 9221   Basecbs 12425   +g cplusg 12489   Grpcgrp 12735   invgcminusg 12736   -gcsg 12737  ._gcmg 12839

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 612  ax-in2 613  ax-io 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-13 2146  ax-14 2147  ax-ext 2155  ax-coll 4110  ax-sep 4113  ax-nul 4121  ax-pow 4166  ax-pr 4200  ax-un 4424  ax-setind 4527  ax-iinf 4578  ax-cnex 7874  ax-resscn 7875  ax-1cn 7876  ax-1re 7877  ax-icn 7878  ax-addcl 7879  ax-addrcl 7880  ax-mulcl 7881  ax-addcom 7883  ax-addass 7885  ax-distr 7887  ax-i2m1 7888  ax-0lt1 7889  ax-0id 7891  ax-rnegex 7892  ax-cnre 7894  ax-pre-ltirr 7895  ax-pre-ltwlin 7896  ax-pre-lttrn 7897  ax-pre-ltadd 7899

This theorem depends on definitions:  df-bi 117  df-dc 833  df-3or 977  df-3an 978  df-tru 1354  df-fal 1357  df-nf 1457  df-sb 1759  df-eu 2025  df-mo 2026  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-ne 2344  df-nel 2439  df-ral 2456  df-rex 2457  df-reu 2458  df-rmo 2459  df-rab 2460  df-v 2735  df-sbc 2959  df-csb 3053  df-dif 3126  df-un 3128  df-in 3130  df-ss 3137  df-nul 3418  df-if 3530  df-pw 3571  df-sn 3592  df-pr 3593  df-op 3595  df-uni 3803  df-int 3838  df-iun 3881  df-br 3996  df-opab 4057  df-mpt 4058  df-tr 4094  df-id 4284  df-iord 4357  df-on 4359  df-ilim 4360  df-suc 4362  df-iom 4581  df-xp 4623  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-res 4629  df-ima 4630  df-iota 5167  df-fun 5207  df-fn 5208  df-f 5209  df-f1 5210  df-fo 5211  df-f1o 5212  df-fv 5213  df-riota 5818  df-ov 5865  df-oprab 5866  df-mpo 5867  df-1st 6128  df-2nd 6129  df-recs 6293  df-frec 6379  df-pnf 7965  df-mnf 7966  df-xr 7967  df-ltxr 7968  df-le 7969  df-sub 8101  df-neg 8102  df-inn 8888  df-2 8946  df-n0 9145  df-z 9222  df-uz 9497  df-fz 9975  df-seqfrec 10411  df-ndx 12428  df-slot 12429  df-base 12431  df-plusg 12502  df-0g 12625  df-mgm 12637  df-sgrp 12670  df-mnd 12680  df-grp 12738  df-minusg 12739  df-sbg 12740  df-mulg 12840

This theorem is referenced by: (None)