Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapcl | Structured version Visualization version GIF version |
Description: Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hgmapcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmapcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmapcl.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmapcl.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmapcl.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hgmapcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hgmapcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
hgmapcl | ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmapcl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hgmapcl.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | eqid 2736 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | eqid 2736 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
5 | hgmapcl.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hgmapcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
7 | eqid 2736 | . . 3 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
8 | eqid 2736 | . . 3 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
9 | eqid 2736 | . . 3 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
10 | hgmapcl.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
11 | hgmapcl.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | hgmapcl.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | hgmapval 40148 | . 2 ⊢ (𝜑 → (𝐺‘𝐹) = (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝐹( ·𝑠 ‘𝑈)𝑥)) = (𝑔( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12 | hdmap14lem15 40143 | . . 3 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝐹( ·𝑠 ‘𝑈)𝑥)) = (𝑔( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) |
15 | riotacl 7304 | . . 3 ⊢ (∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝐹( ·𝑠 ‘𝑈)𝑥)) = (𝑔( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)) → (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝐹( ·𝑠 ‘𝑈)𝑥)) = (𝑔( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ 𝐵) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝐹( ·𝑠 ‘𝑈)𝑥)) = (𝑔( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ 𝐵) |
17 | 13, 16 | eqeltrd 2837 | 1 ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ∃!wreu 3347 ‘cfv 6473 ℩crio 7285 (class class class)co 7329 Basecbs 17001 Scalarcsca 17054 ·𝑠 cvsca 17055 HLchlt 37610 LHypclh 38245 DVecHcdvh 39339 LCDualclcd 39847 HDMapchdma 40053 HGMapchg 40144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-riotaBAD 37213 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-ot 4581 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-undef 8151 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-n0 12327 df-z 12413 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-0g 17241 df-mre 17384 df-mrc 17385 df-acs 17387 df-proset 18102 df-poset 18120 df-plt 18137 df-lub 18153 df-glb 18154 df-join 18155 df-meet 18156 df-p0 18232 df-p1 18233 df-lat 18239 df-clat 18306 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-cntz 19011 df-oppg 19038 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-drng 20087 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lvec 20463 df-lsatoms 37236 df-lshyp 37237 df-lcv 37279 df-lfl 37318 df-lkr 37346 df-ldual 37384 df-oposet 37436 df-ol 37438 df-oml 37439 df-covers 37526 df-ats 37527 df-atl 37558 df-cvlat 37582 df-hlat 37611 df-llines 37759 df-lplanes 37760 df-lvols 37761 df-lines 37762 df-psubsp 37764 df-pmap 37765 df-padd 38057 df-lhyp 38249 df-laut 38250 df-ldil 38365 df-ltrn 38366 df-trl 38420 df-tgrp 39004 df-tendo 39016 df-edring 39018 df-dveca 39264 df-disoa 39290 df-dvech 39340 df-dib 39400 df-dic 39434 df-dih 39490 df-doch 39609 df-djh 39656 df-lcdual 39848 df-mapd 39886 df-hvmap 40018 df-hdmap1 40054 df-hdmap 40055 df-hgmap 40145 |
This theorem is referenced by: hgmapdcl 40151 hgmapvs 40152 hgmapval1 40154 hgmapmul 40156 hdmapglnm2 40172 hdmapinvlem3 40181 hdmapinvlem4 40182 hdmapglem5 40183 hgmapvvlem1 40184 hgmapvvlem2 40185 hdmapglem7 40190 hlhilsrnglem 40218 |
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