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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapdcl | Structured version Visualization version GIF version |
Description: Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hgmapdcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmapdcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmapdcl.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmapdcl.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmapdcl.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hgmapdcl.q | ⊢ 𝑄 = (Scalar‘𝐶) |
hgmapdcl.a | ⊢ 𝐴 = (Base‘𝑄) |
hgmapdcl.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hgmapdcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hgmapdcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
hgmapdcl | ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmapdcl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hgmapdcl.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hgmapdcl.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
4 | hgmapdcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
5 | hgmapdcl.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
6 | hgmapdcl.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | hgmapdcl.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | hgmapcl 37681 | . 2 ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) |
9 | hgmapdcl.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | hgmapdcl.q | . . 3 ⊢ 𝑄 = (Scalar‘𝐶) | |
11 | hgmapdcl.a | . . 3 ⊢ 𝐴 = (Base‘𝑄) | |
12 | 1, 2, 3, 4, 9, 10, 11, 6 | lcdsbase 37389 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
13 | 8, 12 | eleqtrrd 2840 | 1 ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1630 ∈ wcel 2137 ‘cfv 6047 Basecbs 16057 Scalarcsca 16144 HLchlt 35138 LHypclh 35771 DVecHcdvh 36867 LCDualclcd 37375 HGMapchg 37675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 ax-riotaBAD 34740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-ot 4328 df-uni 4587 df-int 4626 df-iun 4672 df-iin 4673 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-of 7060 df-om 7229 df-1st 7331 df-2nd 7332 df-tpos 7519 df-undef 7566 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-er 7909 df-map 8023 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-n0 11483 df-z 11568 df-uz 11878 df-fz 12518 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16154 df-mulr 16155 df-sca 16157 df-vsca 16158 df-0g 16302 df-mre 16446 df-mrc 16447 df-acs 16449 df-preset 17127 df-poset 17145 df-plt 17157 df-lub 17173 df-glb 17174 df-join 17175 df-meet 17176 df-p0 17238 df-p1 17239 df-lat 17245 df-clat 17307 df-mgm 17441 df-sgrp 17483 df-mnd 17494 df-submnd 17535 df-grp 17624 df-minusg 17625 df-sbg 17626 df-subg 17790 df-cntz 17948 df-oppg 17974 df-lsm 18249 df-cmn 18393 df-abl 18394 df-mgp 18688 df-ur 18700 df-ring 18747 df-oppr 18821 df-dvdsr 18839 df-unit 18840 df-invr 18870 df-dvr 18881 df-drng 18949 df-lmod 19065 df-lss 19133 df-lsp 19172 df-lvec 19303 df-lsatoms 34764 df-lshyp 34765 df-lcv 34807 df-lfl 34846 df-lkr 34874 df-ldual 34912 df-oposet 34964 df-ol 34966 df-oml 34967 df-covers 35054 df-ats 35055 df-atl 35086 df-cvlat 35110 df-hlat 35139 df-llines 35285 df-lplanes 35286 df-lvols 35287 df-lines 35288 df-psubsp 35290 df-pmap 35291 df-padd 35583 df-lhyp 35775 df-laut 35776 df-ldil 35891 df-ltrn 35892 df-trl 35947 df-tgrp 36531 df-tendo 36543 df-edring 36545 df-dveca 36791 df-disoa 36818 df-dvech 36868 df-dib 36928 df-dic 36962 df-dih 37018 df-doch 37137 df-djh 37184 df-lcdual 37376 df-mapd 37414 df-hvmap 37546 df-hdmap1 37583 df-hdmap 37584 df-hgmap 37676 |
This theorem is referenced by: hgmapval0 37684 hgmapadd 37686 hgmapmul 37687 hgmaprnlem1N 37688 hgmaprnN 37693 |
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