Theorem ipsubdir 19987

Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
phlsrng.f	|- F = (Scalar ` W )
phllmhm.h	|- ., = ( .i ` W )
phllmhm.v	|- V = ( Base ` W )
ipsubdir.m	|- .- = ( -g ` W )
ipsubdir.s	|- S = ( -g ` F )

Assertion

Ref	Expression
ipsubdir	|- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) ., C ) = ( ( A ., C ) S ( B ., C ) ) )

Proof of Theorem ipsubdir

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> W e. PreHil )
2		phllmod 19975	. . . . . . . 8 |- ( W e. PreHil -> W e. LMod )
3	2	adantr 481	. . . . . . 7 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> W e. LMod )
4		lmodgrp 18870	. . . . . . 7 |- ( W e. LMod -> W e. Grp )
5	3, 4	syl 17	. . . . . 6 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> W e. Grp )
6		simpr1 1067	. . . . . 6 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> A e. V )
7		simpr2 1068	. . . . . 6 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> B e. V )
8		phllmhm.v	. . . . . . 7 |- V = ( Base ` W )
9		ipsubdir.m	. . . . . . 7 |- .- = ( -g ` W )
10	8, 9	grpsubcl 17495	. . . . . 6 |- ( ( W e. Grp /\ A e. V /\ B e. V ) -> ( A .- B ) e. V )
11	5, 6, 7, 10	syl3anc 1326	. . . . 5 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A .- B ) e. V )
12		simpr3 1069	. . . . 5 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
13		phlsrng.f	. . . . . 6 |- F = (Scalar ` W )
14		phllmhm.h	. . . . . 6 |- ., = ( .i ` W )
15		eqid 2622	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
16		eqid 2622	. . . . . 6 |- ( +g ` F ) = ( +g ` F )
17	13, 14, 8, 15, 16	ipdir 19984	. . . . 5 |- ( ( W e. PreHil /\ ( ( A .- B ) e. V /\ B e. V /\ C e. V ) ) -> ( ( ( A .- B ) ( +g ` W ) B ) ., C ) = ( ( ( A .- B ) ., C ) ( +g ` F ) ( B ., C ) ) )
18	1, 11, 7, 12, 17	syl13anc 1328	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( A .- B ) ( +g ` W ) B ) ., C ) = ( ( ( A .- B ) ., C ) ( +g ` F ) ( B ., C ) ) )
19	8, 15, 9	grpnpcan 17507	. . . . . 6 |- ( ( W e. Grp /\ A e. V /\ B e. V ) -> ( ( A .- B ) ( +g ` W ) B ) = A )
20	5, 6, 7, 19	syl3anc 1326	. . . . 5 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) ( +g ` W ) B ) = A )
21	20	oveq1d 6665	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( A .- B ) ( +g ` W ) B ) ., C ) = ( A ., C ) )
22	18, 21	eqtr3d 2658	. . 3 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( A .- B ) ., C ) ( +g ` F ) ( B ., C ) ) = ( A ., C ) )
23	13	lmodfgrp 18872	. . . . 5 |- ( W e. LMod -> F e. Grp )
24	3, 23	syl 17	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> F e. Grp )
25		eqid 2622	. . . . . 6 |- ( Base ` F ) = ( Base ` F )
26	13, 14, 8, 25	ipcl 19978	. . . . 5 |- ( ( W e. PreHil /\ A e. V /\ C e. V ) -> ( A ., C ) e. ( Base ` F ) )
27	1, 6, 12, 26	syl3anc 1326	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., C ) e. ( Base ` F ) )
28	13, 14, 8, 25	ipcl 19978	. . . . 5 |- ( ( W e. PreHil /\ B e. V /\ C e. V ) -> ( B ., C ) e. ( Base ` F ) )
29	1, 7, 12, 28	syl3anc 1326	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( B ., C ) e. ( Base ` F ) )
30	13, 14, 8, 25	ipcl 19978	. . . . 5 |- ( ( W e. PreHil /\ ( A .- B ) e. V /\ C e. V ) -> ( ( A .- B ) ., C ) e. ( Base ` F ) )
31	1, 11, 12, 30	syl3anc 1326	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) ., C ) e. ( Base ` F ) )
32		ipsubdir.s	. . . . 5 |- S = ( -g ` F )
33	25, 16, 32	grpsubadd 17503	. . . 4 |- ( ( F e. Grp /\ ( ( A ., C ) e. ( Base ` F ) /\ ( B ., C ) e. ( Base ` F ) /\ ( ( A .- B ) ., C ) e. ( Base ` F ) ) ) -> ( ( ( A ., C ) S ( B ., C ) ) = ( ( A .- B ) ., C ) <-> ( ( ( A .- B ) ., C ) ( +g ` F ) ( B ., C ) ) = ( A ., C ) ) )
34	24, 27, 29, 31, 33	syl13anc 1328	. . 3 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( A ., C ) S ( B ., C ) ) = ( ( A .- B ) ., C ) <-> ( ( ( A .- B ) ., C ) ( +g ` F ) ( B ., C ) ) = ( A ., C ) ) )
35	22, 34	mpbird 247	. 2 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A ., C ) S ( B ., C ) ) = ( ( A .- B ) ., C ) )
36	35	eqcomd 2628	1 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) ., C ) = ( ( A ., C ) S ( B ., C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .icip 15946   Grpcgrp 17422   -gcsg 17424   LModclmod 18863   PreHilcphl 19969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658  df-ring 18549  df-lmod 18865  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971

This theorem is referenced by:  ip2subdi  19989  cphsubdir  23008