Theorem ip2di 19986

Description: Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
phlsrng.f	|- F = (Scalar ` W )
phllmhm.h	|- ., = ( .i ` W )
phllmhm.v	|- V = ( Base ` W )
ipdir.g	|- .+ = ( +g ` W )
ipdir.p	|- .+^ = ( +g ` F )
ip2di.1	|- ( ph -> W e. PreHil )
ip2di.2	|- ( ph -> A e. V )
ip2di.3	|- ( ph -> B e. V )
ip2di.4	|- ( ph -> C e. V )
ip2di.5	|- ( ph -> D e. V )

Assertion

Ref	Expression
ip2di	|- ( ph -> ( ( A .+ B ) ., ( C .+ D ) ) = ( ( ( A ., C ) .+^ ( B ., D ) ) .+^ ( ( A ., D ) .+^ ( B ., C ) ) ) )

Proof of Theorem ip2di

Step	Hyp	Ref	Expression
1		ip2di.1	. . 3 |- ( ph -> W e. PreHil )
2		ip2di.2	. . 3 |- ( ph -> A e. V )
3		ip2di.3	. . 3 |- ( ph -> B e. V )
4		phllmod 19975	. . . . 5 |- ( W e. PreHil -> W e. LMod )
5	1, 4	syl 17	. . . 4 |- ( ph -> W e. LMod )
6		ip2di.4	. . . 4 |- ( ph -> C e. V )
7		ip2di.5	. . . 4 |- ( ph -> D e. V )
8		phllmhm.v	. . . . 5 |- V = ( Base ` W )
9		ipdir.g	. . . . 5 |- .+ = ( +g ` W )
10	8, 9	lmodvacl 18877	. . . 4 |- ( ( W e. LMod /\ C e. V /\ D e. V ) -> ( C .+ D ) e. V )
11	5, 6, 7, 10	syl3anc 1326	. . 3 |- ( ph -> ( C .+ D ) e. V )
12		phlsrng.f	. . . 4 |- F = (Scalar ` W )
13		phllmhm.h	. . . 4 |- ., = ( .i ` W )
14		ipdir.p	. . . 4 |- .+^ = ( +g ` F )
15	12, 13, 8, 9, 14	ipdir 19984	. . 3 |- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ ( C .+ D ) e. V ) ) -> ( ( A .+ B ) ., ( C .+ D ) ) = ( ( A ., ( C .+ D ) ) .+^ ( B ., ( C .+ D ) ) ) )
16	1, 2, 3, 11, 15	syl13anc 1328	. 2 |- ( ph -> ( ( A .+ B ) ., ( C .+ D ) ) = ( ( A ., ( C .+ D ) ) .+^ ( B ., ( C .+ D ) ) ) )
17	12, 13, 8, 9, 14	ipdi 19985	. . . 4 |- ( ( W e. PreHil /\ ( A e. V /\ C e. V /\ D e. V ) ) -> ( A ., ( C .+ D ) ) = ( ( A ., C ) .+^ ( A ., D ) ) )
18	1, 2, 6, 7, 17	syl13anc 1328	. . 3 |- ( ph -> ( A ., ( C .+ D ) ) = ( ( A ., C ) .+^ ( A ., D ) ) )
19	12, 13, 8, 9, 14	ipdi 19985	. . . . 5 |- ( ( W e. PreHil /\ ( B e. V /\ C e. V /\ D e. V ) ) -> ( B ., ( C .+ D ) ) = ( ( B ., C ) .+^ ( B ., D ) ) )
20	1, 3, 6, 7, 19	syl13anc 1328	. . . 4 |- ( ph -> ( B ., ( C .+ D ) ) = ( ( B ., C ) .+^ ( B ., D ) ) )
21	12	phlsrng 19976	. . . . . 6 |- ( W e. PreHil -> F e. *Ring )
22		srngring 18852	. . . . . 6 |- ( F e. *Ring -> F e. Ring )
23		ringcmn 18581	. . . . . 6 |- ( F e. Ring -> F e. CMnd )
24	1, 21, 22, 23	4syl 19	. . . . 5 |- ( ph -> F e. CMnd )
25		eqid 2622	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
26	12, 13, 8, 25	ipcl 19978	. . . . . 6 |- ( ( W e. PreHil /\ B e. V /\ C e. V ) -> ( B ., C ) e. ( Base ` F ) )
27	1, 3, 6, 26	syl3anc 1326	. . . . 5 |- ( ph -> ( B ., C ) e. ( Base ` F ) )
28	12, 13, 8, 25	ipcl 19978	. . . . . 6 |- ( ( W e. PreHil /\ B e. V /\ D e. V ) -> ( B ., D ) e. ( Base ` F ) )
29	1, 3, 7, 28	syl3anc 1326	. . . . 5 |- ( ph -> ( B ., D ) e. ( Base ` F ) )
30	25, 14	cmncom 18209	. . . . 5 |- ( ( F e. CMnd /\ ( B ., C ) e. ( Base ` F ) /\ ( B ., D ) e. ( Base ` F ) ) -> ( ( B ., C ) .+^ ( B ., D ) ) = ( ( B ., D ) .+^ ( B ., C ) ) )
31	24, 27, 29, 30	syl3anc 1326	. . . 4 |- ( ph -> ( ( B ., C ) .+^ ( B ., D ) ) = ( ( B ., D ) .+^ ( B ., C ) ) )
32	20, 31	eqtrd 2656	. . 3 |- ( ph -> ( B ., ( C .+ D ) ) = ( ( B ., D ) .+^ ( B ., C ) ) )
33	18, 32	oveq12d 6668	. 2 |- ( ph -> ( ( A ., ( C .+ D ) ) .+^ ( B ., ( C .+ D ) ) ) = ( ( ( A ., C ) .+^ ( A ., D ) ) .+^ ( ( B ., D ) .+^ ( B ., C ) ) ) )
34	12, 13, 8, 25	ipcl 19978	. . . 4 |- ( ( W e. PreHil /\ A e. V /\ C e. V ) -> ( A ., C ) e. ( Base ` F ) )
35	1, 2, 6, 34	syl3anc 1326	. . 3 |- ( ph -> ( A ., C ) e. ( Base ` F ) )
36	12, 13, 8, 25	ipcl 19978	. . . 4 |- ( ( W e. PreHil /\ A e. V /\ D e. V ) -> ( A ., D ) e. ( Base ` F ) )
37	1, 2, 7, 36	syl3anc 1326	. . 3 |- ( ph -> ( A ., D ) e. ( Base ` F ) )
38	25, 14	cmn4 18212	. . 3 |- ( ( F e. CMnd /\ ( ( A ., C ) e. ( Base ` F ) /\ ( A ., D ) e. ( Base ` F ) ) /\ ( ( B ., D ) e. ( Base ` F ) /\ ( B ., C ) e. ( Base ` F ) ) ) -> ( ( ( A ., C ) .+^ ( A ., D ) ) .+^ ( ( B ., D ) .+^ ( B ., C ) ) ) = ( ( ( A ., C ) .+^ ( B ., D ) ) .+^ ( ( A ., D ) .+^ ( B ., C ) ) ) )
39	24, 35, 37, 29, 27, 38	syl122anc 1335	. 2 |- ( ph -> ( ( ( A ., C ) .+^ ( A ., D ) ) .+^ ( ( B ., D ) .+^ ( B ., C ) ) ) = ( ( ( A ., C ) .+^ ( B ., D ) ) .+^ ( ( A ., D ) .+^ ( B ., C ) ) ) )
40	16, 33, 39	3eqtrd 2660	1 |- ( ph -> ( ( A .+ B ) ., ( C .+ D ) ) = ( ( ( A ., C ) .+^ ( B ., D ) ) .+^ ( ( A ., D ) .+^ ( B ., C ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .icip 15946  CMndccmn 18193   Ringcrg 18547   *Ringcsr 18844   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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  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-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-rnghom 18715  df-staf 18845  df-srng 18846  df-lmod 18865  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971

This theorem is referenced by:  cph2di  23007