hdmapval3lemN - Mathbox for Norm Megill

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Theorem hdmapval3lemN 37129

Description: Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmapval3.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapval3.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapval3.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapval3.v	⊢ 𝑉 = (Base‘𝑈)
hdmapval3.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmapval3.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapval3.d	⊢ 𝐷 = (Base‘𝐶)
hdmapval3.j	⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmapval3.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmapval3.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapval3.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapval3.te	⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))
hdmapval3lem.t	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
hdmapval3lem.x	⊢ (𝜑 → 𝑥 ∈ 𝑉)
hdmapval3lem.xn	⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇}))

**Assertion**
Ref	Expression
hdmapval3lemN	⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑇⟩))

**Proof of Theorem hdmapval3lemN**
Step	Hyp	Ref	Expression
1		hdmapval3.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapval3.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmapval3.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		eqid 2622	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
5		hdmapval3.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
6		hdmapval3.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		hdmapval3.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
8		eqid 2622	. . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
9		eqid 2622	. . 3 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
10		hdmapval3.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
11		hdmapval3.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		eqid 2622	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
13		hdmapval3.j	. . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
14		eqid 2622	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
15		eqid 2622	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
16		hdmapval3.e	. . . . . . 7 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
17	1, 14, 15, 2, 3, 4, 16, 11	dvheveccl 36401	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
18	1, 2, 3, 4, 6, 7, 12, 13, 11, 17	hvmapcl2 37055	. . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0_g‘𝐶)}))
19	18	eldifad 3586	. . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷)
20	1, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17	mapdhvmap 37058	. . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)}))
21	1, 2, 11	dvhlvec 36398	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
22		hdmapval3lem.x	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
23	17	eldifad 3586	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
24		hdmapval3lem.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
25	24	eldifad 3586	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
26		hdmapval3lem.xn	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇}))
27	3, 5, 21, 22, 23, 25, 26	lspindpi 19132	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}) ∧ (𝑁‘{𝑥}) ≠ (𝑁‘{𝑇})))
28	27	simpld 475	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}))
29	28	necomd 2849	. . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑥}))
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22	hdmap1cl 37094	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩) ∈ 𝐷)
31		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩))
32		eqid 2622	. . . . . 6 ⊢ (-_g‘𝑈) = (-_g‘𝑈)
33		eqid 2622	. . . . . 6 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
34		eqid 2622	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
35	1, 2, 11	dvhlmod 36399	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	3, 34, 5, 35, 23, 25	lspprcl 18978	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑇}) ∈ (LSubSp‘𝑈))
37	3, 4, 34, 35, 36, 22, 26	lssneln0 18952	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
38	1, 2, 3, 32, 4, 5, 6, 7, 33, 8, 9, 10, 11, 17, 19, 37, 30, 29, 20	hdmap1eq 37091	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩) ↔ ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑥})) = ((LSpan‘𝐶)‘{(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩)}) ∧ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{(𝐸(-_g‘𝑈)𝑥)})) = ((LSpan‘𝐶)‘{((𝐽‘𝐸)(-_g‘𝐶)(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩))}))))
39	31, 38	mpbid 222	. . . 4 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑥})) = ((LSpan‘𝐶)‘{(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩)}) ∧ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{(𝐸(-_g‘𝑈)𝑥)})) = ((LSpan‘𝐶)‘{((𝐽‘𝐸)(-_g‘𝐶)(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩))})))
40	39	simpld 475	. . 3 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑥})) = ((LSpan‘𝐶)‘{(𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩)}))
41		hdmapval3.te	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))
42	41	necomd 2849	. . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑇}))
43		hdmapval3.s	. . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
44	3, 5, 35, 23, 25	lspprid1 18997	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑇}))
45	34, 5, 35, 36, 44	lspsnel5a 18996	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑇}))
46	45, 45	unssd 3789	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})) ⊆ (𝑁‘{𝐸, 𝑇}))
47	46, 26	ssneldd 3606	. . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})))
48	1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47	hdmapval2 37124	. . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘⟨𝑥, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩), 𝐸⟩))
49	1, 16, 13, 43, 11	hdmapevec 37127	. . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸))
50	48, 49	eqtr3d 2658	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑥, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩), 𝐸⟩) = (𝐽‘𝐸))
51	3, 5, 35, 23, 25	lspprid2 18998	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝐸, 𝑇}))
52	34, 5, 35, 36, 51	lspsnel5a 18996	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝐸, 𝑇}))
53	45, 52	unssd 3789	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝐸, 𝑇}))
54	53, 26	ssneldd 3606	. . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))
55	1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54	hdmapval2 37124	. . . 4 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝑥, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩), 𝑇⟩))
56	55	eqcomd 2628	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑥, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑥⟩), 𝑇⟩) = (𝑆‘𝑇))
57	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 40, 37, 17, 24, 42, 26, 50, 56	hdmap1eq4N 37096	. 2 ⊢ (𝜑 → (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑇⟩) = (𝑆‘𝑇))
58	57	eqcomd 2628	1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∪ cun 3572 {csn 4177 {cpr 4179 ⟨cop 4183 ⟨cotp 4185 I cid 5023 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 0_gc0g 16100 -_gcsg 17424 LSubSpclss 18932 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 HVMapchvm 37045 HDMap1chdma1 37081 HDMapchdma 37082

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 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 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-undef 7399 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-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hvmap 37046 df-hdmap1 37083 df-hdmap 37084

This theorem is referenced by: hdmapval3N 37130

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