Theorem hdmap1l6d 35778

Description: Lemmma for hdmap1l6 35786. (Contributed by NM, 1-May-2015.)

Hypotheses

Ref	Expression
hdmap1l6.h	|- H = ( LHyp ` K )
hdmap1l6.u	|- U = ( ( DVecH ` K ) ` W )
hdmap1l6.v	|- V = ( Base ` U )
hdmap1l6.p	|- .+ = ( +g ` U )
hdmap1l6.s	|- .- = ( -g ` U )
hdmap1l6c.o	|- .0. = ( 0g ` U )
hdmap1l6.n	|- N = ( LSpan ` U )
hdmap1l6.c	|- C = ( (LCDual ` K ) ` W )
hdmap1l6.d	|- D = ( Base ` C )
hdmap1l6.a	|- .+b = ( +g ` C )
hdmap1l6.r	|- R = ( -g ` C )
hdmap1l6.q	|- Q = ( 0g ` C )
hdmap1l6.l	|- L = ( LSpan ` C )
hdmap1l6.m	|- M = ( (mapd ` K ) ` W )
hdmap1l6.i	|- I = ( (HDMap1 ` K ) ` W )
hdmap1l6.k	|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1l6.f	|- ( ph -> F e. D )
hdmap1l6cl.x	|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1l6.mn	|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1l6d.xn	|- ( ph -> -. X e. ( N ` { Y , Z } ) )
hdmap1l6d.yz	|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
hdmap1l6d.y	|- ( ph -> Y e. ( V \ { .0. } ) )
hdmap1l6d.z	|- ( ph -> Z e. ( V \ { .0. } ) )
hdmap1l6d.w	|- ( ph -> w e. ( V \ { .0. } ) )
hdmap1l6d.wn	|- ( ph -> -. w e. ( N ` { X , Y } ) )

Assertion

Ref	Expression
hdmap1l6d	|- ( ph -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )

Proof of Theorem hdmap1l6d

Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . . . . 6 |- H = ( LHyp ` K )
2		hdmap1l6.c	. . . . . 6 |- C = ( (LCDual ` K ) ` W )
3		hdmap1l6.k	. . . . . 6 |- ( ph -> ( K e. HL /\ W e. H ) )
4	1, 2, 3	lcdlmod 35556	. . . . 5 |- ( ph -> C e. LMod )
5		hdmap1l6.u	. . . . . 6 |- U = ( ( DVecH ` K ) ` W )
6		hdmap1l6.v	. . . . . 6 |- V = ( Base ` U )
7		hdmap1l6c.o	. . . . . 6 |- .0. = ( 0g ` U )
8		hdmap1l6.n	. . . . . 6 |- N = ( LSpan ` U )
9		hdmap1l6.d	. . . . . 6 |- D = ( Base ` C )
10		hdmap1l6.l	. . . . . 6 |- L = ( LSpan ` C )
11		hdmap1l6.m	. . . . . 6 |- M = ( (mapd ` K ) ` W )
12		hdmap1l6.i	. . . . . 6 |- I = ( (HDMap1 ` K ) ` W )
13		hdmap1l6.f	. . . . . 6 |- ( ph -> F e. D )
14		hdmap1l6.mn	. . . . . 6 |- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
15	1, 5, 3	dvhlvec 35073	. . . . . . . . 9 |- ( ph -> U e. LVec )
16		hdmap1l6d.w	. . . . . . . . . 10 |- ( ph -> w e. ( V \ { .0. } ) )
17	16	eldifad 3443	. . . . . . . . 9 |- ( ph -> w e. V )
18		hdmap1l6cl.x	. . . . . . . . . 10 |- ( ph -> X e. ( V \ { .0. } ) )
19	18	eldifad 3443	. . . . . . . . 9 |- ( ph -> X e. V )
20		hdmap1l6d.y	. . . . . . . . . 10 |- ( ph -> Y e. ( V \ { .0. } ) )
21	20	eldifad 3443	. . . . . . . . 9 |- ( ph -> Y e. V )
22		hdmap1l6d.wn	. . . . . . . . 9 |- ( ph -> -. w e. ( N ` { X , Y } ) )
23	6, 8, 15, 17, 19, 21, 22	lspindpi 17331	. . . . . . . 8 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
24	23	simpld 459	. . . . . . 7 |- ( ph -> ( N ` { w } ) =/= ( N ` { X } ) )
25	24	necomd 2720	. . . . . 6 |- ( ph -> ( N ` { X } ) =/= ( N ` { w } ) )
26	1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17	hdmap1cl 35769	. . . . 5 |- ( ph -> ( I ` <. X , F , w >. ) e. D )
27		hdmap1l6.a	. . . . . 6 |- .+b = ( +g ` C )
28		hdmap1l6.q	. . . . . 6 |- Q = ( 0g ` C )
29	9, 27, 28	lmod0vrid 17097	. . . . 5 |- ( ( C e. LMod /\ ( I ` <. X , F , w >. ) e. D ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
30	4, 26, 29	syl2anc 661	. . . 4 |- ( ph -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
31	30	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
32		oteq3 4173	. . . . . 6 |- ( ( Y .+ Z ) = .0. -> <. X , F , ( Y .+ Z ) >. = <. X , F , .0. >. )
33	32	fveq2d 5798	. . . . 5 |- ( ( Y .+ Z ) = .0. -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , .0. >. ) )
34	1, 5, 6, 7, 2, 9, 28, 12, 3, 13, 19	hdmap1val0 35764	. . . . 5 |- ( ph -> ( I ` <. X , F , .0. >. ) = Q )
35	33, 34	sylan9eqr 2515	. . . 4 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = Q )
36	35	oveq2d 6211	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b Q ) )
37		oveq2 6203	. . . . . 6 |- ( ( Y .+ Z ) = .0. -> ( w .+ ( Y .+ Z ) ) = ( w .+ .0. ) )
38	1, 5, 3	dvhlmod 35074	. . . . . . 7 |- ( ph -> U e. LMod )
39		hdmap1l6.p	. . . . . . . 8 |- .+ = ( +g ` U )
40	6, 39, 7	lmod0vrid 17097	. . . . . . 7 |- ( ( U e. LMod /\ w e. V ) -> ( w .+ .0. ) = w )
41	38, 17, 40	syl2anc 661	. . . . . 6 |- ( ph -> ( w .+ .0. ) = w )
42	37, 41	sylan9eqr 2515	. . . . 5 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( w .+ ( Y .+ Z ) ) = w )
43	42	oteq3d 4176	. . . 4 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> <. X , F , ( w .+ ( Y .+ Z ) ) >. = <. X , F , w >. )
44	43	fveq2d 5798	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( I ` <. X , F , w >. ) )
45	31, 36, 44	3eqtr4rd 2504	. 2 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )
46		hdmap1l6.s	. . 3 |- .- = ( -g ` U )
47		hdmap1l6.r	. . 3 |- R = ( -g ` C )
48	3	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( K e. HL /\ W e. H ) )
49	13	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> F e. D )
50	18	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> X e. ( V \ { .0. } ) )
51	14	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
52	16	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> w e. ( V \ { .0. } ) )
53		hdmap1l6d.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
54	53	eldifad 3443	. . . . . 6 |- ( ph -> Z e. V )
55	6, 39	lmodvacl 17080	. . . . . 6 |- ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
56	38, 21, 54, 55	syl3anc 1219	. . . . 5 |- ( ph -> ( Y .+ Z ) e. V )
57	56	anim1i 568	. . . 4 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
58		eldifsn 4103	. . . 4 |- ( ( Y .+ Z ) e. ( V \ { .0. } ) <-> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
59	57, 58	sylibr 212	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( Y .+ Z ) e. ( V \ { .0. } ) )
60		hdmap1l6d.yz	. . . . . . 7 |- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
61		hdmap1l6d.xn	. . . . . . . . 9 |- ( ph -> -. X e. ( N ` { Y , Z } ) )
62	6, 8, 15, 19, 21, 54, 61	lspindpi 17331	. . . . . . . 8 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
63	62	simpld 459	. . . . . . 7 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
64	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp1 35684	. . . . . 6 |- ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) )
65	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp2 35685	. . . . . 6 |- ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )
66	6, 7, 8, 15, 18, 56, 17, 64, 65	lspindp1 17332	. . . . 5 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) /\ -. X e. ( N ` { w , ( Y .+ Z ) } ) ) )
67	66	simprd 463	. . . 4 |- ( ph -> -. X e. ( N ` { w , ( Y .+ Z ) } ) )
68	67	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. X e. ( N ` { w , ( Y .+ Z ) } ) )
69	23	simprd 463	. . . . . . . . 9 |- ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) )
70	6, 7, 8, 15, 16, 21, 69	lspsnne1 17316	. . . . . . . 8 |- ( ph -> -. w e. ( N ` { Y } ) )
71		eqid 2452	. . . . . . . . . 10 |- ( LSSum ` U ) = ( LSSum ` U )
72	6, 8, 71, 38, 21, 54	lsmpr 17288	. . . . . . . . 9 |- ( ph -> ( N ` { Y , Z } ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
73	60	oveq2d 6211	. . . . . . . . 9 |- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
74		eqid 2452	. . . . . . . . . . . . 13 |- ( LSubSp ` U ) = ( LSubSp ` U )
75	6, 74, 8	lspsncl 17176	. . . . . . . . . . . 12 |- ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
76	38, 21, 75	syl2anc 661	. . . . . . . . . . 11 |- ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
77	74	lsssubg 17156	. . . . . . . . . . 11 |- ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) ) -> ( N ` { Y } ) e. (SubGrp ` U ) )
78	38, 76, 77	syl2anc 661	. . . . . . . . . 10 |- ( ph -> ( N ` { Y } ) e. (SubGrp ` U ) )
79	71	lsmidm 16277	. . . . . . . . . 10 |- ( ( N ` { Y } ) e. (SubGrp ` U ) -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) )
80	78, 79	syl 16	. . . . . . . . 9 |- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) )
81	72, 73, 80	3eqtr2d 2499	. . . . . . . 8 |- ( ph -> ( N ` { Y , Z } ) = ( N ` { Y } ) )
82	70, 81	neleqtrrd 2565	. . . . . . 7 |- ( ph -> -. w e. ( N ` { Y , Z } ) )
83	6, 39, 8, 38, 21, 54, 17, 82	lspindp4 17336	. . . . . 6 |- ( ph -> -. w e. ( N ` { Y , ( Y .+ Z ) } ) )
84	6, 8, 15, 17, 21, 56, 83	lspindpi 17331	. . . . 5 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) ) )
85	84	simprd 463	. . . 4 |- ( ph -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) )
86	85	adantr 465	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) )
87		eqidd 2453	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , w >. ) = ( I ` <. X , F , w >. ) )
88		eqidd 2453	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , ( Y .+ Z ) >. ) )
89	1, 5, 6, 39, 46, 7, 8, 2, 9, 27, 47, 28, 10, 11, 12, 48, 49, 50, 51, 52, 59, 68, 86, 87, 88	hdmap1l6a 35774	. 2 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )
90	45, 89	pm2.61dane 2767	1 |- ( ph -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2645   \ cdif 3428   {csn 3980   {cpr 3982   <.cotp 3988   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352   0gc0g 14492   -gcsg 15527  SubGrpcsubg 15789   LSSumclsm 16249   LModclmod 17066   LSubSpclss 17131   LSpanclspn 17170   HLchlt 33314   LHypclh 33947   DVecHcdvh 35042  LCDualclcd 35550  mapdcmpd 35588  HDMap1chdma1 35756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-riotaBAD 32923

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-ot 3989  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-undef 6897  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-0g 14494  df-mre 14638  df-mrc 14639  df-acs 14641  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-mnd 15529  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-subg 15792  df-cntz 15949  df-oppg 15975  df-lsm 16251  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-invr 16882  df-dvr 16893  df-drng 16952  df-lmod 17068  df-lss 17132  df-lsp 17171  df-lvec 17302  df-lsatoms 32940  df-lshyp 32941  df-lcv 32983  df-lfl 33022  df-lkr 33050  df-ldual 33088  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122  df-tgrp 34706  df-tendo 34718  df-edring 34720  df-dveca 34966  df-disoa 34993  df-dvech 35043  df-dib 35103  df-dic 35137  df-dih 35193  df-doch 35312  df-djh 35359  df-lcdual 35551  df-mapd 35589  df-hdmap1 35758

This theorem is referenced by:  hdmap1l6g  35781