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Mirrors > Home > MPE Home > Th. List > Mathboxes > hvmap1o | Structured version Visualization version GIF version |
Description: The vector to functional map provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 27-Mar-2015.) |
Ref | Expression |
---|---|
hvmap1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hvmap1o.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
hvmap1o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hvmap1o.v | ⊢ 𝑉 = (Base‘𝑈) |
hvmap1o.z | ⊢ 0 = (0g‘𝑈) |
hvmap1o.f | ⊢ 𝐹 = (LFnl‘𝑈) |
hvmap1o.l | ⊢ 𝐿 = (LKer‘𝑈) |
hvmap1o.d | ⊢ 𝐷 = (LDual‘𝑈) |
hvmap1o.q | ⊢ 𝑄 = (0g‘𝐷) |
hvmap1o.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
hvmap1o.m | ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) |
hvmap1o.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hvmap1o | ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmap1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hvmap1o.o | . . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
3 | hvmap1o.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hvmap1o.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | eqid 2740 | . . 3 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
6 | eqid 2740 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
7 | eqid 2740 | . . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
8 | eqid 2740 | . . 3 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
9 | hvmap1o.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
10 | hvmap1o.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
11 | hvmap1o.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
12 | hvmap1o.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
13 | hvmap1o.q | . . 3 ⊢ 𝑄 = (0g‘𝐷) | |
14 | hvmap1o.c | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
15 | eqid 2740 | . . 3 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ (𝑂‘{𝑥})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝑥))))) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ (𝑂‘{𝑥})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝑥))))) | |
16 | hvmap1o.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 | lcf1o 39561 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ (𝑂‘{𝑥})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝑥))))):(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄})) |
18 | hvmap1o.m | . . . 4 ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) | |
19 | 1, 3, 2, 4, 5, 6, 9, 7, 8, 18, 16 | hvmapfval 39769 | . . 3 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ (𝑂‘{𝑥})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝑥)))))) |
20 | 19 | f1oeq1d 6709 | . 2 ⊢ (𝜑 → (𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}) ↔ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ (𝑂‘{𝑥})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝑥))))):(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))) |
21 | 17, 20 | mpbird 256 | 1 ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 {crab 3070 ∖ cdif 3889 {csn 4567 ↦ cmpt 5162 –1-1-onto→wf1o 6431 ‘cfv 6432 ℩crio 7227 (class class class)co 7271 Basecbs 16910 +gcplusg 16960 Scalarcsca 16963 ·𝑠 cvsca 16964 0gc0g 17148 LFnlclfn 37067 LKerclk 37095 LDualcld 37133 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 ocHcoch 39357 HVMapchvm 39766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-lshyp 36987 df-lfl 37068 df-lkr 37096 df-ldual 37134 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tgrp 38753 df-tendo 38765 df-edring 38767 df-dveca 39013 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 df-doch 39358 df-djh 39405 df-hvmap 39767 |
This theorem is referenced by: hvmapclN 39774 hvmap1o2 39775 |
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