Theorem ldualvsubval 34444

Description: The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 34442? (Requires D to opp_r conversion.) (Contributed by NM, 26-Feb-2015.)

Hypotheses

Ref	Expression
ldualvsubval.v	|- V = ( Base ` W )
ldualvsubval.r	|- R = (Scalar ` W )
ldualvsubval.s	|- S = ( -g ` R )
ldualvsubval.f	|- F = (LFnl ` W )
ldualvsubval.d	|- D = (LDual ` W )
ldualvsubval.m	|- .- = ( -g ` D )
ldualvsubval.w	|- ( ph -> W e. LMod )
ldualvsubval.g	|- ( ph -> G e. F )
ldualvsubval.h	|- ( ph -> H e. F )
ldualvsubval.x	|- ( ph -> X e. V )

Assertion

Ref	Expression
ldualvsubval	|- ( ph -> ( ( G .- H ) ` X ) = ( ( G ` X ) S ( H ` X ) ) )

Proof of Theorem ldualvsubval

Step	Hyp	Ref	Expression
1		ldualvsubval.d	. . . . 5 |- D = (LDual ` W )
2		ldualvsubval.w	. . . . 5 |- ( ph -> W e. LMod )
3	1, 2	lduallmod 34440	. . . 4 |- ( ph -> D e. LMod )
4		ldualvsubval.f	. . . . 5 |- F = (LFnl ` W )
5		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
6		ldualvsubval.g	. . . . 5 |- ( ph -> G e. F )
7	4, 1, 5, 2, 6	ldualelvbase 34414	. . . 4 |- ( ph -> G e. ( Base ` D ) )
8		ldualvsubval.h	. . . . 5 |- ( ph -> H e. F )
9	4, 1, 5, 2, 8	ldualelvbase 34414	. . . 4 |- ( ph -> H e. ( Base ` D ) )
10		eqid 2622	. . . . 5 |- ( +g ` D ) = ( +g ` D )
11		ldualvsubval.m	. . . . 5 |- .- = ( -g ` D )
12		eqid 2622	. . . . 5 |- (Scalar ` D ) = (Scalar ` D )
13		eqid 2622	. . . . 5 |- ( .s ` D ) = ( .s ` D )
14		eqid 2622	. . . . 5 |- ( invg ` (Scalar ` D ) ) = ( invg ` (Scalar ` D ) )
15		eqid 2622	. . . . 5 |- ( 1r ` (Scalar ` D ) ) = ( 1r ` (Scalar ` D ) )
16	5, 10, 11, 12, 13, 14, 15	lmodvsubval2 18918	. . . 4 |- ( ( D e. LMod /\ G e. ( Base ` D ) /\ H e. ( Base ` D ) ) -> ( G .- H ) = ( G ( +g ` D ) ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ) )
17	3, 7, 9, 16	syl3anc 1326	. . 3 |- ( ph -> ( G .- H ) = ( G ( +g ` D ) ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ) )
18	17	fveq1d 6193	. 2 |- ( ph -> ( ( G .- H ) ` X ) = ( ( G ( +g ` D ) ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ) ` X ) )
19		ldualvsubval.v	. . 3 |- V = ( Base ` W )
20		ldualvsubval.r	. . 3 |- R = (Scalar ` W )
21		eqid 2622	. . 3 |- ( +g ` R ) = ( +g ` R )
22		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
23	12	lmodfgrp 18872	. . . . . . 7 |- ( D e. LMod -> (Scalar ` D ) e. Grp )
24	3, 23	syl 17	. . . . . 6 |- ( ph -> (Scalar ` D ) e. Grp )
25	12	lmodring 18871	. . . . . . . 8 |- ( D e. LMod -> (Scalar ` D ) e. Ring )
26	3, 25	syl 17	. . . . . . 7 |- ( ph -> (Scalar ` D ) e. Ring )
27		eqid 2622	. . . . . . . 8 |- ( Base ` (Scalar ` D ) ) = ( Base ` (Scalar ` D ) )
28	27, 15	ringidcl 18568	. . . . . . 7 |- ( (Scalar ` D ) e. Ring -> ( 1r ` (Scalar ` D ) ) e. ( Base ` (Scalar ` D ) ) )
29	26, 28	syl 17	. . . . . 6 |- ( ph -> ( 1r ` (Scalar ` D ) ) e. ( Base ` (Scalar ` D ) ) )
30	27, 14	grpinvcl 17467	. . . . . 6 |- ( ( (Scalar ` D ) e. Grp /\ ( 1r ` (Scalar ` D ) ) e. ( Base ` (Scalar ` D ) ) ) -> ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) e. ( Base ` (Scalar ` D ) ) )
31	24, 29, 30	syl2anc 693	. . . . 5 |- ( ph -> ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) e. ( Base ` (Scalar ` D ) ) )
32	20, 22, 1, 12, 27, 2	ldualsbase 34420	. . . . 5 |- ( ph -> ( Base ` (Scalar ` D ) ) = ( Base ` R ) )
33	31, 32	eleqtrd 2703	. . . 4 |- ( ph -> ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) e. ( Base ` R ) )
34	4, 20, 22, 1, 13, 2, 33, 8	ldualvscl 34426	. . 3 |- ( ph -> ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) e. F )
35		ldualvsubval.x	. . 3 |- ( ph -> X e. V )
36	19, 20, 21, 4, 1, 10, 2, 6, 34, 35	ldualvaddval 34418	. 2 |- ( ph -> ( ( G ( +g ` D ) ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ) ` X ) = ( ( G ` X ) ( +g ` R ) ( ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ` X ) ) )
37		eqid 2622	. . . . . . . . 9 |- ( invg ` R ) = ( invg ` R )
38	20, 37, 1, 12, 14, 2	ldualneg 34436	. . . . . . . 8 |- ( ph -> ( invg ` (Scalar ` D ) ) = ( invg ` R ) )
39		eqid 2622	. . . . . . . . 9 |- ( 1r ` R ) = ( 1r ` R )
40	20, 39, 1, 12, 15, 2	ldual1 34435	. . . . . . . 8 |- ( ph -> ( 1r ` (Scalar ` D ) ) = ( 1r ` R ) )
41	38, 40	fveq12d 6197	. . . . . . 7 |- ( ph -> ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
42	41	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) = ( ( ( invg ` R ) ` ( 1r ` R ) ) ( .s ` D ) H ) )
43	42	fveq1d 6193	. . . . 5 |- ( ph -> ( ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ` X ) = ( ( ( ( invg ` R ) ` ( 1r ` R ) ) ( .s ` D ) H ) ` X ) )
44		eqid 2622	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
45	20	lmodring 18871	. . . . . . . . 9 |- ( W e. LMod -> R e. Ring )
46	2, 45	syl 17	. . . . . . . 8 |- ( ph -> R e. Ring )
47		ringgrp 18552	. . . . . . . 8 |- ( R e. Ring -> R e. Grp )
48	46, 47	syl 17	. . . . . . 7 |- ( ph -> R e. Grp )
49	20, 22, 39	lmod1cl 18890	. . . . . . . 8 |- ( W e. LMod -> ( 1r ` R ) e. ( Base ` R ) )
50	2, 49	syl 17	. . . . . . 7 |- ( ph -> ( 1r ` R ) e. ( Base ` R ) )
51	22, 37	grpinvcl 17467	. . . . . . 7 |- ( ( R e. Grp /\ ( 1r ` R ) e. ( Base ` R ) ) -> ( ( invg ` R ) ` ( 1r ` R ) ) e. ( Base ` R ) )
52	48, 50, 51	syl2anc 693	. . . . . 6 |- ( ph -> ( ( invg ` R ) ` ( 1r ` R ) ) e. ( Base ` R ) )
53	4, 19, 20, 22, 44, 1, 13, 2, 52, 8, 35	ldualvsval 34425	. . . . 5 |- ( ph -> ( ( ( ( invg ` R ) ` ( 1r ` R ) ) ( .s ` D ) H ) ` X ) = ( ( H ` X ) ( .r ` R ) ( ( invg ` R ) ` ( 1r ` R ) ) ) )
54	20, 22, 19, 4	lflcl 34351	. . . . . . 7 |- ( ( W e. LMod /\ H e. F /\ X e. V ) -> ( H ` X ) e. ( Base ` R ) )
55	2, 8, 35, 54	syl3anc 1326	. . . . . 6 |- ( ph -> ( H ` X ) e. ( Base ` R ) )
56	22, 44, 39, 37, 46, 55	rngnegr 18595	. . . . 5 |- ( ph -> ( ( H ` X ) ( .r ` R ) ( ( invg ` R ) ` ( 1r ` R ) ) ) = ( ( invg ` R ) ` ( H ` X ) ) )
57	43, 53, 56	3eqtrd 2660	. . . 4 |- ( ph -> ( ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ` X ) = ( ( invg ` R ) ` ( H ` X ) ) )
58	57	oveq2d 6666	. . 3 |- ( ph -> ( ( G ` X ) ( +g ` R ) ( ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ` X ) ) = ( ( G ` X ) ( +g ` R ) ( ( invg ` R ) ` ( H ` X ) ) ) )
59	20, 22, 19, 4	lflcl 34351	. . . . 5 |- ( ( W e. LMod /\ G e. F /\ X e. V ) -> ( G ` X ) e. ( Base ` R ) )
60	2, 6, 35, 59	syl3anc 1326	. . . 4 |- ( ph -> ( G ` X ) e. ( Base ` R ) )
61		ldualvsubval.s	. . . . 5 |- S = ( -g ` R )
62	22, 21, 37, 61	grpsubval 17465	. . . 4 |- ( ( ( G ` X ) e. ( Base ` R ) /\ ( H ` X ) e. ( Base ` R ) ) -> ( ( G ` X ) S ( H ` X ) ) = ( ( G ` X ) ( +g ` R ) ( ( invg ` R ) ` ( H ` X ) ) ) )
63	60, 55, 62	syl2anc 693	. . 3 |- ( ph -> ( ( G ` X ) S ( H ` X ) ) = ( ( G ` X ) ( +g ` R ) ( ( invg ` R ) ` ( H ` X ) ) ) )
64	58, 63	eqtr4d 2659	. 2 |- ( ph -> ( ( G ` X ) ( +g ` R ) ( ( ( ( invg ` (Scalar ` D ) ) ` ( 1r ` (Scalar ` D ) ) ) ( .s ` D ) H ) ` X ) ) = ( ( G ` X ) S ( H ` X ) ) )
65	18, 36, 64	3eqtrd 2660	1 |- ( ph -> ( ( G .- H ) ` X ) = ( ( G ` X ) S ( H ` X ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   1rcur 18501   Ringcrg 18547   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-tpos 7352  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-mulr 15955  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-oppr 18623  df-lmod 18865  df-lfl 34345  df-ldual 34411

This theorem is referenced by:  lcfrlem1  36831