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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapcl | Structured version Visualization version GIF version |
Description: Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.) |
Ref | Expression |
---|---|
hdmapcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapcl.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapcl.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmapcl.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmapcl.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapcl.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmapcl | ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapcl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
3 | hdmapcl.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hdmapcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | eqid 2798 | . . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
6 | hdmapcl.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | hdmapcl.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
8 | eqid 2798 | . . 3 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊) | |
9 | eqid 2798 | . . 3 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊) | |
10 | hdmapcl.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
11 | hdmapcl.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | hdmapcl.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | hdmapval 39124 | . 2 ⊢ (𝜑 → (𝑆‘𝑇) = (℩ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘〈𝑦, (((HDMap1‘𝐾)‘𝑊)‘〈〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉, (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉), 𝑦〉), 𝑇〉)))) |
14 | eqid 2798 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
15 | eqid 2798 | . . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
16 | eqid 2798 | . . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
17 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | eqid 2798 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
19 | 1, 17, 18, 3, 4, 14, 2, 11 | dvheveccl 38408 | . . . . 5 ⊢ (𝜑 → 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
20 | 1, 3, 4, 14, 5, 6, 15, 16, 8, 11, 19 | mapdhvmap 39065 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉})) = ((LSpan‘𝐶)‘{(((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉)})) |
21 | eqid 2798 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
22 | 1, 3, 4, 14, 6, 7, 21, 8, 11, 19 | hvmapcl2 39062 | . . . . 5 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
23 | 22 | eldifad 3893 | . . . 4 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉) ∈ 𝐷) |
24 | 1, 3, 4, 14, 5, 6, 7, 15, 16, 9, 11, 20, 19, 23, 12 | hdmap1eu 39120 | . . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘〈𝑦, (((HDMap1‘𝐾)‘𝑊)‘〈〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉, (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉), 𝑦〉), 𝑇〉))) |
25 | riotacl 7110 | . . 3 ⊢ (∃!ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘〈𝑦, (((HDMap1‘𝐾)‘𝑊)‘〈〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉, (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉), 𝑦〉), 𝑇〉)) → (℩ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘〈𝑦, (((HDMap1‘𝐾)‘𝑊)‘〈〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉, (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉), 𝑦〉), 𝑇〉))) ∈ 𝐷) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘〈𝑦, (((HDMap1‘𝐾)‘𝑊)‘〈〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉, (((HVMap‘𝐾)‘𝑊)‘〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉), 𝑦〉), 𝑇〉))) ∈ 𝐷) |
27 | 13, 26 | eqeltrd 2890 | 1 ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃!wreu 3108 ∪ cun 3879 {csn 4525 〈cop 4531 〈cotp 4533 I cid 5424 ↾ cres 5521 ‘cfv 6324 ℩crio 7092 Basecbs 16475 0gc0g 16705 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920 HVMapchvm 39052 HDMap1chdma1 39087 HDMapchdma 39088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921 df-hvmap 39053 df-hdmap1 39089 df-hdmap 39090 |
This theorem is referenced by: hdmapval2 39128 hdmap10lem 39135 hdmapeq0 39140 hdmapnzcl 39141 hdmapneg 39142 hdmapsub 39143 hdmap11 39144 hdmaprnlem3N 39146 hdmaprnlem3uN 39147 hdmaprnlem7N 39151 hdmaprnlem8N 39152 hdmaprnlem9N 39153 hdmaprnlem3eN 39154 hdmaprnN 39160 hdmap14lem2a 39163 hdmap14lem2N 39165 hdmap14lem3 39166 hdmap14lem4a 39167 hdmap14lem6 39169 hdmap14lem8 39171 hgmapval0 39188 hgmapval1 39189 hgmapadd 39190 hgmapmul 39191 hgmaprnlem1N 39192 hgmaprnlem2N 39193 hgmaprnlem4N 39195 hdmapipcl 39201 hdmapln1 39202 hdmaplna1 39203 hdmaplns1 39204 hdmaplnm1 39205 hdmaplna2 39206 hdmapglnm2 39207 hdmaplkr 39209 hdmapellkr 39210 hdmapip0 39211 hdmapinvlem1 39214 hdmapinvlem3 39216 |
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