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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdspex | Structured version Visualization version GIF version |
Description: The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.) |
Ref | Expression |
---|---|
mapdspex.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdspex.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdspex.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdspex.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdspex.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdspex.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdspex.b | ⊢ 𝐵 = (Base‘𝐶) |
mapdspex.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdspex.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdspex.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
mapdspex | ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdspex.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdspex.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | mapdspex.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 37741 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | 4 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → 𝐶 ∈ LMod) |
6 | mapdspex.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
7 | mapdspex.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2777 | . . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
9 | eqid 2777 | . . . 4 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶) | |
10 | 3 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) | |
12 | 1, 6, 7, 8, 2, 9, 10, 11 | mapdat 37816 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝑀‘(𝑁‘{𝑋})) ∈ (LSAtoms‘𝐶)) |
13 | mapdspex.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
14 | mapdspex.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
15 | 13, 14, 9 | islsati 35143 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑋})) ∈ (LSAtoms‘𝐶)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
16 | 5, 12, 15 | syl2anc 579 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
17 | eqid 2777 | . . . . 5 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
18 | 1, 2, 13, 17, 3 | lcd0vcl 37763 | . . . 4 ⊢ (𝜑 → (0g‘𝐶) ∈ 𝐵) |
19 | 18 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (0g‘𝐶) ∈ 𝐵) |
20 | fveq2 6446 | . . . 4 ⊢ ((𝑁‘{𝑋}) = {(0g‘𝑈)} → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{(0g‘𝑈)})) | |
21 | eqid 2777 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
22 | 1, 6, 7, 21, 2, 17, 3 | mapd0 37814 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
23 | 17, 14 | lspsn0 19403 | . . . . . 6 ⊢ (𝐶 ∈ LMod → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
24 | 4, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
25 | 22, 24 | eqtr4d 2816 | . . . 4 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = (𝐽‘{(0g‘𝐶)})) |
26 | 20, 25 | sylan9eqr 2835 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) |
27 | sneq 4407 | . . . . 5 ⊢ (𝑔 = (0g‘𝐶) → {𝑔} = {(0g‘𝐶)}) | |
28 | 27 | fveq2d 6450 | . . . 4 ⊢ (𝑔 = (0g‘𝐶) → (𝐽‘{𝑔}) = (𝐽‘{(0g‘𝐶)})) |
29 | 28 | rspceeqv 3528 | . . 3 ⊢ (((0g‘𝐶) ∈ 𝐵 ∧ (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
30 | 19, 26, 29 | syl2anc 579 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
31 | mapdspex.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
32 | mapdspex.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
33 | 1, 7, 3 | dvhlmod 37259 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
34 | mapdspex.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
35 | 31, 32, 21, 8, 33, 34 | lsator0sp 35150 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈) ∨ (𝑁‘{𝑋}) = {(0g‘𝑈)})) |
36 | 16, 30, 35 | mpjaodan 944 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∃wrex 3090 {csn 4397 ‘cfv 6135 Basecbs 16255 0gc0g 16486 LModclmod 19255 LSpanclspn 19366 LSAtomsclsa 35123 HLchlt 35499 LHypclh 36133 DVecHcdvh 37227 LCDualclcd 37735 mapdcmpd 37773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35102 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35125 df-lshyp 35126 df-lcv 35168 df-lfl 35207 df-lkr 35235 df-ldual 35273 df-oposet 35325 df-ol 35327 df-oml 35328 df-covers 35415 df-ats 35416 df-atl 35447 df-cvlat 35471 df-hlat 35500 df-llines 35647 df-lplanes 35648 df-lvols 35649 df-lines 35650 df-psubsp 35652 df-pmap 35653 df-padd 35945 df-lhyp 36137 df-laut 36138 df-ldil 36253 df-ltrn 36254 df-trl 36308 df-tgrp 36892 df-tendo 36904 df-edring 36906 df-dveca 37152 df-disoa 37178 df-dvech 37228 df-dib 37288 df-dic 37322 df-dih 37378 df-doch 37497 df-djh 37544 df-lcdual 37736 df-mapd 37774 |
This theorem is referenced by: mapdpglem2 37822 |
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