Theorem hdmap1l6d 36611

Description: Lemmma for hdmap1l6 36619. (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 36389	. . . . 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 35906	. . . . . . . . 9 |- ( ph -> U e. LVec )
16		hdmap1l6d.w	. . . . . . . . . 10 |- ( ph -> w e. ( V \ { .0. } ) )
17	16	eldifad 3488	. . . . . . . . 9 |- ( ph -> w e. V )
18		hdmap1l6cl.x	. . . . . . . . . 10 |- ( ph -> X e. ( V \ { .0. } ) )
19	18	eldifad 3488	. . . . . . . . 9 |- ( ph -> X e. V )
20		hdmap1l6d.y	. . . . . . . . . 10 |- ( ph -> Y e. ( V \ { .0. } ) )
21	20	eldifad 3488	. . . . . . . . 9 |- ( ph -> Y e. V )
22		hdmap1l6d.wn	. . . . . . . . 9 |- ( ph -> -. w e. ( N ` { X , Y } ) )
23	6, 8, 15, 17, 19, 21, 22	lspindpi 17561	. . . . . . . 8 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
24	23	simpld 459	. . . . . . 7 |- ( ph -> ( N ` { w } ) =/= ( N ` { X } ) )
25	24	necomd 2738	. . . . . 6 |- ( ph -> ( N ` { X } ) =/= ( N ` { w } ) )
26	1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17	hdmap1cl 36602	. . . . 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 17326	. . . . 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 4224	. . . . . 6 |- ( ( Y .+ Z ) = .0. -> <. X , F , ( Y .+ Z ) >. = <. X , F , .0. >. )
33	32	fveq2d 5868	. . . . 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 36597	. . . . 5 |- ( ph -> ( I ` <. X , F , .0. >. ) = Q )
35	33, 34	sylan9eqr 2530	. . . 4 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = Q )
36	35	oveq2d 6298	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b Q ) )
37		oveq2 6290	. . . . . 6 |- ( ( Y .+ Z ) = .0. -> ( w .+ ( Y .+ Z ) ) = ( w .+ .0. ) )
38	1, 5, 3	dvhlmod 35907	. . . . . . 7 |- ( ph -> U e. LMod )
39		hdmap1l6.p	. . . . . . . 8 |- .+ = ( +g ` U )
40	6, 39, 7	lmod0vrid 17326	. . . . . . 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 2530	. . . . 5 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( w .+ ( Y .+ Z ) ) = w )
43	42	oteq3d 4227	. . . 4 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> <. X , F , ( w .+ ( Y .+ Z ) ) >. = <. X , F , w >. )
44	43	fveq2d 5868	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( I ` <. X , F , w >. ) )
45	31, 36, 44	3eqtr4rd 2519	. 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 3488	. . . . . 6 |- ( ph -> Z e. V )
55	6, 39	lmodvacl 17309	. . . . . 6 |- ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
56	38, 21, 54, 55	syl3anc 1228	. . . . 5 |- ( ph -> ( Y .+ Z ) e. V )
57	56	anim1i 568	. . . 4 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
58		eldifsn 4152	. . . 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 17561	. . . . . . . 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 36517	. . . . . 6 |- ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) )
65	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp2 36518	. . . . . 6 |- ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )
66	6, 7, 8, 15, 18, 56, 17, 64, 65	lspindp1 17562	. . . . 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 17546	. . . . . . . 8 |- ( ph -> -. w e. ( N ` { Y } ) )
71		eqid 2467	. . . . . . . . . 10 |- ( LSSum ` U ) = ( LSSum ` U )
72	6, 8, 71, 38, 21, 54	lsmpr 17518	. . . . . . . . 9 |- ( ph -> ( N ` { Y , Z } ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
73	60	oveq2d 6298	. . . . . . . . 9 |- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
74		eqid 2467	. . . . . . . . . . . . 13 |- ( LSubSp ` U ) = ( LSubSp ` U )
75	6, 74, 8	lspsncl 17406	. . . . . . . . . . . 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 17386	. . . . . . . . . . 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 16478	. . . . . . . . . 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 2514	. . . . . . . 8 |- ( ph -> ( N ` { Y , Z } ) = ( N ` { Y } ) )
82	70, 81	neleqtrrd 2580	. . . . . . 7 |- ( ph -> -. w e. ( N ` { Y , Z } ) )
83	6, 39, 8, 38, 21, 54, 17, 82	lspindp4 17566	. . . . . 6 |- ( ph -> -. w e. ( N ` { Y , ( Y .+ Z ) } ) )
84	6, 8, 15, 17, 21, 56, 83	lspindpi 17561	. . . . 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 2468	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , w >. ) = ( I ` <. X , F , w >. ) )
88		eqidd 2468	. . 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 36607	. 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 2785	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 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   {csn 4027   {cpr 4029   <.cotp 4035   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   -gcsg 15726  SubGrpcsubg 15990   LSSumclsm 16450   LModclmod 17295   LSubSpclss 17361   LSpanclspn 17400   HLchlt 34147   LHypclh 34780   DVecHcdvh 35875  LCDualclcd 36383  mapdcmpd 36421  HDMap1chdma1 36589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-0g 14693  df-mre 14837  df-mrc 14838  df-acs 14840  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-cntz 16150  df-oppg 16176  df-lsm 16452  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-dvr 17116  df-drng 17181  df-lmod 17297  df-lss 17362  df-lsp 17401  df-lvec 17532  df-lsatoms 33773  df-lshyp 33774  df-lcv 33816  df-lfl 33855  df-lkr 33883  df-ldual 33921  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tgrp 35539  df-tendo 35551  df-edring 35553  df-dveca 35799  df-disoa 35826  df-dvech 35876  df-dib 35936  df-dic 35970  df-dih 36026  df-doch 36145  df-djh 36192  df-lcdual 36384  df-mapd 36422  df-hdmap1 36591

This theorem is referenced by:  hdmap1l6g  36614