Theorem ldualvsdi1 34430

Description: Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)

Hypotheses

Ref	Expression
ldualvsdi1.f	|- F = (LFnl ` W )
ldualvsdi1.r	|- R = (Scalar ` W )
ldualvsdi1.k	|- K = ( Base ` R )
ldualvsdi1.d	|- D = (LDual ` W )
ldualvsdi1.p	|- .+ = ( +g ` D )
ldualvsdi1.s	|- .x. = ( .s ` D )
ldualvsdi1.w	|- ( ph -> W e. LMod )
ldualvsdi1.x	|- ( ph -> X e. K )
ldualvsdi1.g	|- ( ph -> G e. F )
ldualvsdi1.h	|- ( ph -> H e. F )

Assertion

Ref	Expression
ldualvsdi1	|- ( ph -> ( X .x. ( G .+ H ) ) = ( ( X .x. G ) .+ ( X .x. H ) ) )

Proof of Theorem ldualvsdi1

Step	Hyp	Ref	Expression
1		ldualvsdi1.f	. . . 4 |- F = (LFnl ` W )
2		eqid 2622	. . . 4 |- ( Base ` W ) = ( Base ` W )
3		ldualvsdi1.r	. . . 4 |- R = (Scalar ` W )
4		ldualvsdi1.k	. . . 4 |- K = ( Base ` R )
5		eqid 2622	. . . 4 |- ( .r ` R ) = ( .r ` R )
6		ldualvsdi1.d	. . . 4 |- D = (LDual ` W )
7		ldualvsdi1.s	. . . 4 |- .x. = ( .s ` D )
8		ldualvsdi1.w	. . . 4 |- ( ph -> W e. LMod )
9		ldualvsdi1.x	. . . 4 |- ( ph -> X e. K )
10		ldualvsdi1.g	. . . 4 |- ( ph -> G e. F )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ldualvs 34424	. . 3 |- ( ph -> ( X .x. G ) = ( G oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) )
12		ldualvsdi1.h	. . . 4 |- ( ph -> H e. F )
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	ldualvs 34424	. . 3 |- ( ph -> ( X .x. H ) = ( H oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) )
14	11, 13	oveq12d 6668	. 2 |- ( ph -> ( ( X .x. G ) oF ( +g ` R ) ( X .x. H ) ) = ( ( G oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) oF ( +g ` R ) ( H oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) ) )
15		eqid 2622	. . 3 |- ( +g ` R ) = ( +g ` R )
16		ldualvsdi1.p	. . 3 |- .+ = ( +g ` D )
17	1, 3, 4, 6, 7, 8, 9, 10	ldualvscl 34426	. . 3 |- ( ph -> ( X .x. G ) e. F )
18	1, 3, 4, 6, 7, 8, 9, 12	ldualvscl 34426	. . 3 |- ( ph -> ( X .x. H ) e. F )
19	1, 3, 15, 6, 16, 8, 17, 18	ldualvadd 34416	. 2 |- ( ph -> ( ( X .x. G ) .+ ( X .x. H ) ) = ( ( X .x. G ) oF ( +g ` R ) ( X .x. H ) ) )
20	1, 6, 16, 8, 10, 12	ldualvaddcl 34417	. . . 4 |- ( ph -> ( G .+ H ) e. F )
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 20	ldualvs 34424	. . 3 |- ( ph -> ( X .x. ( G .+ H ) ) = ( ( G .+ H ) oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) )
22	1, 3, 15, 6, 16, 8, 10, 12	ldualvadd 34416	. . . 4 |- ( ph -> ( G .+ H ) = ( G oF ( +g ` R ) H ) )
23	22	oveq1d 6665	. . 3 |- ( ph -> ( ( G .+ H ) oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) = ( ( G oF ( +g ` R ) H ) oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) )
24	2, 3, 4, 15, 5, 1, 8, 9, 10, 12	lflvsdi1 34365	. . 3 |- ( ph -> ( ( G oF ( +g ` R ) H ) oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) = ( ( G oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) oF ( +g ` R ) ( H oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) ) )
25	21, 23, 24	3eqtrd 2660	. 2 |- ( ph -> ( X .x. ( G .+ H ) ) = ( ( G oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) oF ( +g ` R ) ( H oF ( .r ` R ) ( ( Base ` W ) X. { X } ) ) ) )
26	14, 19, 25	3eqtr4rd 2667	1 |- ( ph -> ( X .x. ( G .+ H ) ) = ( ( X .x. G ) .+ ( X .x. H ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   LModclmod 18863  LFnlclfn 34344  LDualcld 34410

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-int 4476  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lfl 34345  df-ldual 34411

This theorem is referenced by:  lduallmodlem  34439