Theorem hdmap1l6d 35091

Description: Lemmma for hdmap1l6 35099. (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 34869	. . . . 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 34386	. . . . . . . . 9 |- ( ph -> U e. LVec )
16		hdmap1l6d.w	. . . . . . . . . 10 |- ( ph -> w e. ( V \ { .0. } ) )
17	16	eldifad 3454	. . . . . . . . 9 |- ( ph -> w e. V )
18		hdmap1l6cl.x	. . . . . . . . . 10 |- ( ph -> X e. ( V \ { .0. } ) )
19	18	eldifad 3454	. . . . . . . . 9 |- ( ph -> X e. V )
20		hdmap1l6d.y	. . . . . . . . . 10 |- ( ph -> Y e. ( V \ { .0. } ) )
21	20	eldifad 3454	. . . . . . . . 9 |- ( ph -> Y e. V )
22		hdmap1l6d.wn	. . . . . . . . 9 |- ( ph -> -. w e. ( N ` { X , Y } ) )
23	6, 8, 15, 17, 19, 21, 22	lspindpi 18290	. . . . . . . 8 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
24	23	simpld 460	. . . . . . 7 |- ( ph -> ( N ` { w } ) =/= ( N ` { X } ) )
25	24	necomd 2702	. . . . . 6 |- ( ph -> ( N ` { X } ) =/= ( N ` { w } ) )
26	1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17	hdmap1cl 35082	. . . . 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 18057	. . . . 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 665	. . . 4 |- ( ph -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
31	30	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
32		oteq3 4201	. . . . . 6 |- ( ( Y .+ Z ) = .0. -> <. X , F , ( Y .+ Z ) >. = <. X , F , .0. >. )
33	32	fveq2d 5885	. . . . 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 35077	. . . . 5 |- ( ph -> ( I ` <. X , F , .0. >. ) = Q )
35	33, 34	sylan9eqr 2492	. . . 4 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = Q )
36	35	oveq2d 6321	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b Q ) )
37		oveq2 6313	. . . . . 6 |- ( ( Y .+ Z ) = .0. -> ( w .+ ( Y .+ Z ) ) = ( w .+ .0. ) )
38	1, 5, 3	dvhlmod 34387	. . . . . . 7 |- ( ph -> U e. LMod )
39		hdmap1l6.p	. . . . . . . 8 |- .+ = ( +g ` U )
40	6, 39, 7	lmod0vrid 18057	. . . . . . 7 |- ( ( U e. LMod /\ w e. V ) -> ( w .+ .0. ) = w )
41	38, 17, 40	syl2anc 665	. . . . . 6 |- ( ph -> ( w .+ .0. ) = w )
42	37, 41	sylan9eqr 2492	. . . . 5 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( w .+ ( Y .+ Z ) ) = w )
43	42	oteq3d 4204	. . . 4 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> <. X , F , ( w .+ ( Y .+ Z ) ) >. = <. X , F , w >. )
44	43	fveq2d 5885	. . 3 |- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( I ` <. X , F , w >. ) )
45	31, 36, 44	3eqtr4rd 2481	. 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 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( K e. HL /\ W e. H ) )
49	13	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> F e. D )
50	18	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> X e. ( V \ { .0. } ) )
51	14	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
52	16	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> w e. ( V \ { .0. } ) )
53		hdmap1l6d.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
54	53	eldifad 3454	. . . . . 6 |- ( ph -> Z e. V )
55	6, 39	lmodvacl 18040	. . . . . 6 |- ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
56	38, 21, 54, 55	syl3anc 1264	. . . . 5 |- ( ph -> ( Y .+ Z ) e. V )
57	56	anim1i 570	. . . 4 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
58		eldifsn 4128	. . . 4 |- ( ( Y .+ Z ) e. ( V \ { .0. } ) <-> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
59	57, 58	sylibr 215	. . 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 18290	. . . . . . . 8 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
63	62	simpld 460	. . . . . . 7 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
64	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp1 34997	. . . . . 6 |- ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) )
65	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp2 34998	. . . . . 6 |- ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )
66	6, 7, 8, 15, 18, 56, 17, 64, 65	lspindp1 18291	. . . . 5 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) /\ -. X e. ( N ` { w , ( Y .+ Z ) } ) ) )
67	66	simprd 464	. . . 4 |- ( ph -> -. X e. ( N ` { w , ( Y .+ Z ) } ) )
68	67	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. X e. ( N ` { w , ( Y .+ Z ) } ) )
69	23	simprd 464	. . . . . . . . 9 |- ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) )
70	6, 7, 8, 15, 16, 21, 69	lspsnne1 18275	. . . . . . . 8 |- ( ph -> -. w e. ( N ` { Y } ) )
71		eqid 2429	. . . . . . . . . 10 |- ( LSSum ` U ) = ( LSSum ` U )
72	6, 8, 71, 38, 21, 54	lsmpr 18247	. . . . . . . . 9 |- ( ph -> ( N ` { Y , Z } ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
73	60	oveq2d 6321	. . . . . . . . 9 |- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
74		eqid 2429	. . . . . . . . . . . . 13 |- ( LSubSp ` U ) = ( LSubSp ` U )
75	6, 74, 8	lspsncl 18135	. . . . . . . . . . . 12 |- ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
76	38, 21, 75	syl2anc 665	. . . . . . . . . . 11 |- ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
77	74	lsssubg 18115	. . . . . . . . . . 11 |- ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) ) -> ( N ` { Y } ) e. (SubGrp ` U ) )
78	38, 76, 77	syl2anc 665	. . . . . . . . . 10 |- ( ph -> ( N ` { Y } ) e. (SubGrp ` U ) )
79	71	lsmidm 17249	. . . . . . . . . 10 |- ( ( N ` { Y } ) e. (SubGrp ` U ) -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) )
80	78, 79	syl 17	. . . . . . . . 9 |- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) )
81	72, 73, 80	3eqtr2d 2476	. . . . . . . 8 |- ( ph -> ( N ` { Y , Z } ) = ( N ` { Y } ) )
82	70, 81	neleqtrrd 2542	. . . . . . 7 |- ( ph -> -. w e. ( N ` { Y , Z } ) )
83	6, 39, 8, 38, 21, 54, 17, 82	lspindp4 18295	. . . . . 6 |- ( ph -> -. w e. ( N ` { Y , ( Y .+ Z ) } ) )
84	6, 8, 15, 17, 21, 56, 83	lspindpi 18290	. . . . 5 |- ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) ) )
85	84	simprd 464	. . . 4 |- ( ph -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) )
86	85	adantr 466	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) )
87		eqidd 2430	. . 3 |- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , w >. ) = ( I ` <. X , F , w >. ) )
88		eqidd 2430	. . 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 35087	. 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 2749	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 370   = wceq 1437   e. wcel 1870   =/= wne 2625   \ cdif 3439   {csn 4002   {cpr 4004   <.cotp 4010   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   0gc0g 15297   -gcsg 16622  SubGrpcsubg 16762   LSSumclsm 17221   LModclmod 18026   LSubSpclss 18090   LSpanclspn 18129   HLchlt 32625   LHypclh 33258   DVecHcdvh 34355  LCDualclcd 34863  mapdcmpd 34901  HDMap1chdma1 35069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32234

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-oppg 16948  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-lsatoms 32251  df-lshyp 32252  df-lcv 32294  df-lfl 32333  df-lkr 32361  df-ldual 32399  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tgrp 34019  df-tendo 34031  df-edring 34033  df-dveca 34279  df-disoa 34306  df-dvech 34356  df-dib 34416  df-dic 34450  df-dih 34506  df-doch 34625  df-djh 34672  df-lcdual 34864  df-mapd 34902  df-hdmap1 35071

This theorem is referenced by:  hdmap1l6g  35094