Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochflcl | Structured version Visualization version GIF version |
Description: Closure of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkrcl 36246. (Contributed by NM, 27-Oct-2014.) |
Ref | Expression |
---|---|
dochflcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochflcl.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochflcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochflcl.v | ⊢ 𝑉 = (Base‘𝑈) |
dochflcl.z | ⊢ 0 = (0g‘𝑈) |
dochflcl.a | ⊢ + = (+g‘𝑈) |
dochflcl.t | ⊢ · = ( ·𝑠 ‘𝑈) |
dochflcl.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochflcl.d | ⊢ 𝐷 = (Scalar‘𝑈) |
dochflcl.r | ⊢ 𝑅 = (Base‘𝐷) |
dochflcl.g | ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) |
dochflcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochflcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
dochflcl | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochflcl.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
2 | dochflcl.a | . 2 ⊢ + = (+g‘𝑈) | |
3 | eqid 2821 | . 2 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
4 | eqid 2821 | . 2 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
5 | eqid 2821 | . 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
6 | dochflcl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | dochflcl.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | dochflcl.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 6, 7, 8 | dvhlvec 38239 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
10 | dochflcl.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
11 | dochflcl.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
12 | dochflcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
13 | 6, 10, 7, 1, 11, 5, 8, 12 | dochsnshp 38583 | . 2 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈)) |
14 | 12 | eldifad 3948 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
15 | 6, 10, 7, 1, 11, 3, 4, 8, 12 | dochexmidat 38589 | . 2 ⊢ (𝜑 → (( ⊥ ‘{𝑋})(LSSum‘𝑈)((LSpan‘𝑈)‘{𝑋})) = 𝑉) |
16 | dochflcl.d | . 2 ⊢ 𝐷 = (Scalar‘𝑈) | |
17 | dochflcl.r | . 2 ⊢ 𝑅 = (Base‘𝐷) | |
18 | dochflcl.t | . 2 ⊢ · = ( ·𝑠 ‘𝑈) | |
19 | dochflcl.g | . 2 ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) | |
20 | dochflcl.f | . 2 ⊢ 𝐹 = (LFnl‘𝑈) | |
21 | 1, 2, 3, 4, 5, 9, 13, 14, 15, 16, 17, 18, 19, 20 | lshpkrcl 36246 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ∖ cdif 3933 {csn 4561 ↦ cmpt 5139 ‘cfv 6350 ℩crio 7107 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 LSSumclsm 18753 LSpanclspn 19737 LSHypclsh 36105 LFnlclfn 36187 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 ocHcoch 38477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lfl 36188 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 df-tendo 37885 df-edring 37887 df-dveca 38133 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 df-djh 38525 |
This theorem is referenced by: lcfl6lem 38628 lcfl7lem 38629 lcfrlem9 38680 lcfrlem10 38682 |
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