dvheveccl - Mathbox for Norm Megill

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Theorem dvheveccl 40585

Description: Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 40589 and dihpN 40809. (Contributed by NM, 27-Mar-2015.)

**Hypotheses**
Ref	Expression
dvheveccl.h	⊢ 𝐻 = (LHyp‘𝐾)
dvheveccl.b	⊢ 𝐵 = (Base‘𝐾)
dvheveccl.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvheveccl.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvheveccl.v	⊢ 𝑉 = (Base‘𝑈)
dvheveccl.z	⊢ 0 = (0_g‘𝑈)
dvheveccl.e	⊢ 𝐸 = ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩
dvheveccl.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
dvheveccl	⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 }))

Proof of Theorem dvheveccl Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvheveccl.e	. 2 ⊢ 𝐸 = ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩
2		dvheveccl.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		dvheveccl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
4		dvheveccl.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
5		dvheveccl.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	3, 4, 5	idltrn 39623	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
7	2, 6	syl 17	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ 𝑇)
8		eqid 2728	. . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
9	4, 5, 8	tendoidcl 40242	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
10	2, 9	syl 17	. . . 4 ⊢ (𝜑 → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
11		dvheveccl.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
12		dvheveccl.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
13	4, 5, 8, 11, 12	dvhelvbasei 40561	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ∈ 𝑉)
14	2, 7, 10, 13	syl12anc 836	. . 3 ⊢ (𝜑 → ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ∈ 𝑉)
15		eqid 2728	. . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
16	3, 4, 5, 8, 15	tendo1ne0 40301	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)))
17	2, 16	syl 17	. . . 4 ⊢ (𝜑 → ( I ↾ 𝑇) ≠ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)))
18		dvheveccl.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
19	3, 4, 5, 11, 18, 15	dvh0g 40584	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩)
20	2, 19	syl 17	. . . . . 6 ⊢ (𝜑 → 0 = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩)
21		eqtr 2751	. . . . . . 7 ⊢ ((⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = 0 ∧ 0 = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩) → ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩)
22		opthg 5479	. . . . . . . . 9 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩ ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)))))
23	7, 10, 22	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩ ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)))))
24		simpr 484	. . . . . . . 8 ⊢ ((( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))) → ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)))
25	23, 24	biimtrdi 252	. . . . . . 7 ⊢ (𝜑 → (⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩ → ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))))
26	21, 25	syl5 34	. . . . . 6 ⊢ (𝜑 → ((⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = 0 ∧ 0 = ⟨( I ↾ 𝐵), (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))⟩) → ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))))
27	20, 26	mpan2d 693	. . . . 5 ⊢ (𝜑 → (⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ = 0 → ( I ↾ 𝑇) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))))
28	27	necon3d 2958	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑇) ≠ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) → ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ≠ 0 ))
29	17, 28	mpd 15	. . 3 ⊢ (𝜑 → ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ≠ 0 )
30		eldifsn 4791	. . 3 ⊢ (⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ∈ (𝑉 ∖ { 0 }) ↔ (⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ∈ 𝑉 ∧ ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ≠ 0 ))
31	14, 29, 30	sylanbrc 582	. 2 ⊢ (𝜑 → ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩ ∈ (𝑉 ∖ { 0 }))
32	1, 31	eqeltrid 2833	1 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 {csn 4629 ⟨cop 4635 ↦ cmpt 5231 I cid 5575 ↾ cres 5680 ‘cfv 6548 Basecbs 17180 0_gc0g 17421 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 TEndoctendo 40225 DVecHcdvh 40551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-riotaBAD 38425

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-0g 17423 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-drng 20626 df-lmod 20745 df-lvec 20988 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tendo 40228 df-edring 40230 df-dvech 40552

This theorem is referenced by: hdmapcl 41303 hdmapval2lem 41304 hdmapval0 41306 hdmapeveclem 41307 hdmapevec 41308 hdmapevec2 41309 hdmapval3lemN 41310 hdmapval3N 41311 hdmap10lem 41312 hdmap11lem1 41314 hdmap11lem2 41315 hdmapinvlem1 41391 hdmapinvlem2 41392 hdmapinvlem3 41393 hdmapinvlem4 41394 hdmapglem5 41395 hgmapvvlem3 41398 hdmapglem7a 41400 hdmapglem7b 41401 hdmapglem7 41402

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