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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 37383 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | 4 | adantr 472 | . . 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 2760 | . . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
9 | eqid 2760 | . . . 4 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶) | |
10 | 3 | adantr 472 | . . . 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 37458 | . . 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 34784 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑋})) ∈ (LSAtoms‘𝐶)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
16 | 5, 12, 15 | syl2anc 696 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
17 | eqid 2760 | . . . . 5 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
18 | 1, 2, 13, 17, 3 | lcd0vcl 37405 | . . . 4 ⊢ (𝜑 → (0g‘𝐶) ∈ 𝐵) |
19 | 18 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (0g‘𝐶) ∈ 𝐵) |
20 | fveq2 6352 | . . . 4 ⊢ ((𝑁‘{𝑋}) = {(0g‘𝑈)} → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{(0g‘𝑈)})) | |
21 | eqid 2760 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
22 | 1, 6, 7, 21, 2, 17, 3 | mapd0 37456 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
23 | 17, 14 | lspsn0 19210 | . . . . . 6 ⊢ (𝐶 ∈ LMod → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
24 | 4, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
25 | 22, 24 | eqtr4d 2797 | . . . 4 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = (𝐽‘{(0g‘𝐶)})) |
26 | 20, 25 | sylan9eqr 2816 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) |
27 | sneq 4331 | . . . . . 6 ⊢ (𝑔 = (0g‘𝐶) → {𝑔} = {(0g‘𝐶)}) | |
28 | 27 | fveq2d 6356 | . . . . 5 ⊢ (𝑔 = (0g‘𝐶) → (𝐽‘{𝑔}) = (𝐽‘{(0g‘𝐶)})) |
29 | 28 | eqeq2d 2770 | . . . 4 ⊢ (𝑔 = (0g‘𝐶) → ((𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔}) ↔ (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)}))) |
30 | 29 | rspcev 3449 | . . 3 ⊢ (((0g‘𝐶) ∈ 𝐵 ∧ (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
31 | 19, 26, 30 | syl2anc 696 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
32 | mapdspex.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
33 | mapdspex.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
34 | 1, 7, 3 | dvhlmod 36901 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
35 | mapdspex.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
36 | 32, 33, 21, 8, 34, 35 | lsator0sp 34791 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈) ∨ (𝑁‘{𝑋}) = {(0g‘𝑈)})) |
37 | 16, 31, 36 | mpjaodan 862 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∃wrex 3051 {csn 4321 ‘cfv 6049 Basecbs 16059 0gc0g 16302 LModclmod 19065 LSpanclspn 19173 LSAtomsclsa 34764 HLchlt 35140 LHypclh 35773 DVecHcdvh 36869 LCDualclcd 37377 mapdcmpd 37415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-riotaBAD 34742 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-tpos 7521 df-undef 7568 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-0g 16304 df-mre 16448 df-mrc 16449 df-acs 16451 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-cntz 17950 df-oppg 17976 df-lsm 18251 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-drng 18951 df-lmod 19067 df-lss 19135 df-lsp 19174 df-lvec 19305 df-lsatoms 34766 df-lshyp 34767 df-lcv 34809 df-lfl 34848 df-lkr 34876 df-ldual 34914 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 df-lvols 35289 df-lines 35290 df-psubsp 35292 df-pmap 35293 df-padd 35585 df-lhyp 35777 df-laut 35778 df-ldil 35893 df-ltrn 35894 df-trl 35949 df-tgrp 36533 df-tendo 36545 df-edring 36547 df-dveca 36793 df-disoa 36820 df-dvech 36870 df-dib 36930 df-dic 36964 df-dih 37020 df-doch 37139 df-djh 37186 df-lcdual 37378 df-mapd 37416 |
This theorem is referenced by: mapdpglem2 37464 |
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