![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 41574 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | 4 | adantr 480 | . . 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 2734 | . . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
9 | eqid 2734 | . . . 4 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶) | |
10 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) | |
12 | 1, 6, 7, 8, 2, 9, 10, 11 | mapdat 41649 | . . 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 38975 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑋})) ∈ (LSAtoms‘𝐶)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
16 | 5, 12, 15 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
17 | eqid 2734 | . . . . 5 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
18 | 1, 2, 13, 17, 3 | lcd0vcl 41596 | . . . 4 ⊢ (𝜑 → (0g‘𝐶) ∈ 𝐵) |
19 | 18 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (0g‘𝐶) ∈ 𝐵) |
20 | fveq2 6906 | . . . 4 ⊢ ((𝑁‘{𝑋}) = {(0g‘𝑈)} → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{(0g‘𝑈)})) | |
21 | eqid 2734 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
22 | 1, 6, 7, 21, 2, 17, 3 | mapd0 41647 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
23 | 17, 14 | lspsn0 21023 | . . . . . 6 ⊢ (𝐶 ∈ LMod → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
24 | 4, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
25 | 22, 24 | eqtr4d 2777 | . . . 4 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = (𝐽‘{(0g‘𝐶)})) |
26 | 20, 25 | sylan9eqr 2796 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) |
27 | sneq 4640 | . . . . 5 ⊢ (𝑔 = (0g‘𝐶) → {𝑔} = {(0g‘𝐶)}) | |
28 | 27 | fveq2d 6910 | . . . 4 ⊢ (𝑔 = (0g‘𝐶) → (𝐽‘{𝑔}) = (𝐽‘{(0g‘𝐶)})) |
29 | 28 | rspceeqv 3644 | . . 3 ⊢ (((0g‘𝐶) ∈ 𝐵 ∧ (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
30 | 19, 26, 29 | syl2anc 584 | . 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 41092 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
34 | mapdspex.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
35 | 31, 32, 21, 8, 33, 34 | lsator0sp 38982 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈) ∨ (𝑁‘{𝑋}) = {(0g‘𝑈)})) |
36 | 16, 30, 35 | mpjaodan 960 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {csn 4630 ‘cfv 6562 Basecbs 17244 0gc0g 17485 LModclmod 20874 LSpanclspn 20986 LSAtomsclsa 38955 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 LCDualclcd 41568 mapdcmpd 41606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-nzr 20529 df-rlreg 20710 df-domn 20711 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lshyp 38958 df-lcv 39000 df-lfl 39039 df-lkr 39067 df-ldual 39105 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tgrp 40725 df-tendo 40737 df-edring 40739 df-dveca 40985 df-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 df-djh 41377 df-lcdual 41569 df-mapd 41607 |
This theorem is referenced by: mapdpglem2 41655 |
Copyright terms: Public domain | W3C validator |