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Mirrors > Home > MPE Home > Th. List > Mathboxes > hvmap1o2 | 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 |
---|---|
hvmap1o2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hvmap1o2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hvmap1o2.v | ⊢ 𝑉 = (Base‘𝑈) |
hvmap1o2.z | ⊢ 0 = (0g‘𝑈) |
hvmap1o2.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hvmap1o2.f | ⊢ 𝐹 = (Base‘𝐶) |
hvmap1o2.o | ⊢ 𝑂 = (0g‘𝐶) |
hvmap1o2.m | ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) |
hvmap1o2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hvmap1o2 | ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmap1o2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
3 | hvmap1o2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hvmap1o2.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | hvmap1o2.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | eqid 2821 | . . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2821 | . . 3 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2821 | . . 3 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | eqid 2821 | . . 3 ⊢ (0g‘(LDual‘𝑈)) = (0g‘(LDual‘𝑈)) | |
10 | eqid 2821 | . . 3 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} | |
11 | hvmap1o2.m | . . 3 ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) | |
12 | hvmap1o2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | hvmap1o 38893 | . 2 ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∖ {(0g‘(LDual‘𝑈))})) |
14 | hvmap1o2.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
15 | hvmap1o2.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
16 | 1, 2, 14, 15, 3, 6, 7, 10, 12 | lcdvbase 38723 | . . . 4 ⊢ (𝜑 → 𝐹 = {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) |
17 | hvmap1o2.o | . . . . . 6 ⊢ 𝑂 = (0g‘𝐶) | |
18 | 1, 3, 8, 9, 14, 17, 12 | lcd0v2 38742 | . . . . 5 ⊢ (𝜑 → 𝑂 = (0g‘(LDual‘𝑈))) |
19 | 18 | sneqd 4572 | . . . 4 ⊢ (𝜑 → {𝑂} = {(0g‘(LDual‘𝑈))}) |
20 | 16, 19 | difeq12d 4099 | . . 3 ⊢ (𝜑 → (𝐹 ∖ {𝑂}) = ({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∖ {(0g‘(LDual‘𝑈))})) |
21 | f1oeq3 6600 | . . 3 ⊢ ((𝐹 ∖ {𝑂}) = ({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∖ {(0g‘(LDual‘𝑈))}) → (𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂}) ↔ 𝑀:(𝑉 ∖ { 0 })–1-1-onto→({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∖ {(0g‘(LDual‘𝑈))}))) | |
22 | 20, 21 | syl 17 | . 2 ⊢ (𝜑 → (𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂}) ↔ 𝑀:(𝑉 ∖ { 0 })–1-1-onto→({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∖ {(0g‘(LDual‘𝑈))}))) |
23 | 13, 22 | mpbird 259 | 1 ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ∖ cdif 3932 {csn 4560 –1-1-onto→wf1o 6348 ‘cfv 6349 Basecbs 16477 0gc0g 16707 LFnlclfn 36187 LKerclk 36215 LDualcld 36253 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 ocHcoch 38477 LCDualclcd 38716 HVMapchvm 38886 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 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-mre 16851 df-mrc 16852 df-acs 16854 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-oppg 18468 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-lcv 36149 df-lfl 36188 df-lkr 36216 df-ldual 36254 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 df-lcdual 38717 df-hvmap 38887 |
This theorem is referenced by: hvmapcl2 38896 |
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