hdmapevec - Mathbox for Norm Megill

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Theorem hdmapevec 39828

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 2739	. . 3 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
3		eqid 2739	. . 3 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
4		eqid 2739	. . 3 ⊢ (LSpan‘((DVecH‘𝐾)‘𝑊)) = (LSpan‘((DVecH‘𝐾)‘𝑊))
5		hdmapevec.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		eqid 2739	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		eqid 2739	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
8		eqid 2739	. . . . 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 39105	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((Base‘((DVecH‘𝐾)‘𝑊)) ∖ {(0_g‘((DVecH‘𝐾)‘𝑊))}))
11	10	eldifad 3903	. . 3 ⊢ (𝜑 → 𝐸 ∈ (Base‘((DVecH‘𝐾)‘𝑊)))
12	1, 2, 3, 4, 5, 11	dvh2dim 39438	. 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 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		eqid 2739	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
17		eqid 2739	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
18		eqid 2739	. . . 4 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
19		simp2 1135	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)))
20		ssid 3947	. . . . . . . 8 ⊢ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ⊆ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})
21	20, 20	unssi 4123	. . . . . . 7 ⊢ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) ⊆ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})
22	21	sseli 3921	. . . . . 6 ⊢ (𝑧 ∈ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}))
23	22	con3i 154	. . . . 5 ⊢ (¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) → ¬ 𝑧 ∈ (((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸}) ∪ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})))
24	23	3ad2ant3 1133	. . . 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 39827	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ¬ 𝑧 ∈ ((LSpan‘((DVecH‘𝐾)‘𝑊))‘{𝐸})) → (𝑆‘𝐸) = (𝐽‘𝐸))
26	25	rexlimdv3a 3216	. 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 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∃wrex 3066 ∪ cun 3889 {csn 4566 ⟨cop 4572 I cid 5487 ↾ cres 5590 ‘cfv 6430 Basecbs 16893 0_gc0g 17131 LSpanclspn 20214 HLchlt 37343 LHypclh 37977 LTrncltrn 38094 DVecHcdvh 39071 LCDualclcd 39579 HVMapchvm 39749 HDMap1chdma1 39784 HDMapchdma 39785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-riotaBAD 36946

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-undef 8073 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-0g 17133 df-mre 17276 df-mrc 17277 df-acs 17279 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-join 18047 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-clat 18198 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-cntz 18904 df-oppg 18931 df-lsm 19222 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 df-lsatoms 36969 df-lshyp 36970 df-lcv 37012 df-lfl 37051 df-lkr 37079 df-ldual 37117 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-llines 37491 df-lplanes 37492 df-lvols 37493 df-lines 37494 df-psubsp 37496 df-pmap 37497 df-padd 37789 df-lhyp 37981 df-laut 37982 df-ldil 38097 df-ltrn 38098 df-trl 38152 df-tgrp 38736 df-tendo 38748 df-edring 38750 df-dveca 38996 df-disoa 39022 df-dvech 39072 df-dib 39132 df-dic 39166 df-dih 39222 df-doch 39341 df-djh 39388 df-lcdual 39580 df-mapd 39618 df-hvmap 39750 df-hdmap1 39786 df-hdmap 39787

This theorem is referenced by: hdmapevec2 39829 hdmapval3lemN 39830

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