hdmapevec - Mathbox for Norm Megill

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Theorem hdmapevec 38449

Description: Value of map from vectors to functionals at the reference vector 𝐸. (Contributed by NM, 16-May-2015.)

**Hypotheses**
Ref	Expression
hdmapevec.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapevec.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapevec.j	⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmapevec.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapevec.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
hdmapevec	⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸))

Proof of Theorem hdmapevec Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapevec.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2773	. . 3 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
3		eqid 2773	. . 3 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
4		eqid 2773	. . 3 ⊢ (LSpan‘((DVecH‘𝐾)‘𝑊)) = (LSpan‘((DVecH‘𝐾)‘𝑊))
5		hdmapevec.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		eqid 2773	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		eqid 2773	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
8		eqid 2773	. . . . 5 ⊢ (0_g‘((DVecH‘𝐾)‘𝑊)) = (0_g‘((DVecH‘𝐾)‘𝑊))
9		hdmapevec.e	. . . . 5 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
10	1, 6, 7, 2, 3, 8, 9, 5	dvheveccl 37726	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((Base‘((DVecH‘𝐾)‘𝑊)) ∖ {(0_g‘((DVecH‘𝐾)‘𝑊))}))
11	10	eldifad 3836	. . 3 ⊢ (𝜑 → 𝐸 ∈ (Base‘((DVecH‘𝐾)‘𝑊)))
12	1, 2, 3, 4, 5, 11	dvh2dim 38059	. 2 ⊢ (𝜑 → ∃𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}))
13		hdmapevec.j	. . . 4 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
14		hdmapevec.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
15	5	3ad2ant1 1114	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		eqid 2773	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
17		eqid 2773	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
18		eqid 2773	. . . 4 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
19		simp2 1118	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)))
20		ssid 3874	. . . . . . . 8 ⊢ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ⊆ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})
21	20, 20	unssi 4044	. . . . . . 7 ⊢ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) ⊆ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})
22	21	sseli 3849	. . . . . 6 ⊢ (𝑧 ∈ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}))
23	22	con3i 152	. . . . 5 ⊢ (¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) → ¬ 𝑧 ∈ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})))
24	23	3ad2ant3 1116	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → ¬ 𝑧 ∈ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})))
25	1, 9, 13, 14, 15, 2, 3, 4, 16, 17, 18, 19, 24	hdmapeveclem 38448	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → (𝑆‘𝐸) = (𝐽‘𝐸))
26	25	rexlimdv3a 3226	. 2 ⊢ (𝜑 → (∃𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) → (𝑆‘𝐸) = (𝐽‘𝐸)))
27	12, 26	mpd 15	1 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∃wrex 3084 ∪ cun 3822 {csn 4436 ⟨cop 4442 I cid 5308 ↾ cres 5406 ‘cfv 6186 Basecbs 16338 0_gc0g 16568 LSpanclspn 19478 HLchlt 35964 LHypclh 36598 LTrncltrn 36715 DVecHcdvh 37692 LCDualclcd 38200 HVMapchvm 38370 HDMap1chdma1 38405 HDMapchdma 38406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-riotaBAD 35567

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-ot 4445 df-uni 4710 df-int 4747 df-iun 4791 df-iin 4792 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-om 7396 df-1st 7500 df-2nd 7501 df-tpos 7694 df-undef 7741 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-sca 16436 df-vsca 16437 df-0g 16570 df-mre 16728 df-mrc 16729 df-acs 16731 df-proset 17409 df-poset 17427 df-plt 17439 df-lub 17455 df-glb 17456 df-join 17457 df-meet 17458 df-p0 17520 df-p1 17521 df-lat 17527 df-clat 17589 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-submnd 17817 df-grp 17907 df-minusg 17908 df-sbg 17909 df-subg 18073 df-cntz 18231 df-oppg 18258 df-lsm 18535 df-cmn 18681 df-abl 18682 df-mgp 18976 df-ur 18988 df-ring 19035 df-oppr 19109 df-dvdsr 19127 df-unit 19128 df-invr 19158 df-dvr 19169 df-drng 19240 df-lmod 19371 df-lss 19439 df-lsp 19479 df-lvec 19610 df-lsatoms 35590 df-lshyp 35591 df-lcv 35633 df-lfl 35672 df-lkr 35700 df-ldual 35738 df-oposet 35790 df-ol 35792 df-oml 35793 df-covers 35880 df-ats 35881 df-atl 35912 df-cvlat 35936 df-hlat 35965 df-llines 36112 df-lplanes 36113 df-lvols 36114 df-lines 36115 df-psubsp 36117 df-pmap 36118 df-padd 36410 df-lhyp 36602 df-laut 36603 df-ldil 36718 df-ltrn 36719 df-trl 36773 df-tgrp 37357 df-tendo 37369 df-edring 37371 df-dveca 37617 df-disoa 37643 df-dvech 37693 df-dib 37753 df-dic 37787 df-dih 37843 df-doch 37962 df-djh 38009 df-lcdual 38201 df-mapd 38239 df-hvmap 38371 df-hdmap1 38407 df-hdmap 38408

This theorem is referenced by: hdmapevec2 38450 hdmapval3lemN 38451

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