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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 38722 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | 4 | adantr 483 | . . 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 2821 | . . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
9 | eqid 2821 | . . . 4 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶) | |
10 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) | |
12 | 1, 6, 7, 8, 2, 9, 10, 11 | mapdat 38797 | . . 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 36124 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑋})) ∈ (LSAtoms‘𝐶)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
16 | 5, 12, 15 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈)) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
17 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
18 | 1, 2, 13, 17, 3 | lcd0vcl 38744 | . . . 4 ⊢ (𝜑 → (0g‘𝐶) ∈ 𝐵) |
19 | 18 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (0g‘𝐶) ∈ 𝐵) |
20 | fveq2 6665 | . . . 4 ⊢ ((𝑁‘{𝑋}) = {(0g‘𝑈)} → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{(0g‘𝑈)})) | |
21 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
22 | 1, 6, 7, 21, 2, 17, 3 | mapd0 38795 | . . . . 5 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
23 | 17, 14 | lspsn0 19774 | . . . . . 6 ⊢ (𝐶 ∈ LMod → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
24 | 4, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐽‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
25 | 22, 24 | eqtr4d 2859 | . . . 4 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = (𝐽‘{(0g‘𝐶)})) |
26 | 20, 25 | sylan9eqr 2878 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) = {(0g‘𝑈)}) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) |
27 | sneq 4571 | . . . . 5 ⊢ (𝑔 = (0g‘𝐶) → {𝑔} = {(0g‘𝐶)}) | |
28 | 27 | fveq2d 6669 | . . . 4 ⊢ (𝑔 = (0g‘𝐶) → (𝐽‘{𝑔}) = (𝐽‘{(0g‘𝐶)})) |
29 | 28 | rspceeqv 3638 | . . 3 ⊢ (((0g‘𝐶) ∈ 𝐵 ∧ (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(0g‘𝐶)})) → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
30 | 19, 26, 29 | syl2anc 586 | . 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 38240 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
34 | mapdspex.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
35 | 31, 32, 21, 8, 33, 34 | lsator0sp 36131 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ (LSAtoms‘𝑈) ∨ (𝑁‘{𝑋}) = {(0g‘𝑈)})) |
36 | 16, 30, 35 | mpjaodan 955 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {csn 4561 ‘cfv 6350 Basecbs 16477 0gc0g 16707 LModclmod 19628 LSpanclspn 19737 LSAtomsclsa 36104 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 LCDualclcd 38716 mapdcmpd 38754 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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-mapd 38755 |
This theorem is referenced by: mapdpglem2 38803 |
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