mapdrval - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdrval		Structured version Visualization version GIF version

Theorem mapdrval 41606

Description: Given a dual subspace 𝑅 (of functionals with closed kernels), reconstruct the subspace 𝑄 that maps to it. (Contributed by NM, 12-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdrval.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdrval.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdrval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdrval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdrval.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdrval.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdrval.l	⊢ 𝐿 = (LKer‘𝑈)
mapdrval.d	⊢ 𝐷 = (LDual‘𝑈)
mapdrval.t	⊢ 𝑇 = (LSubSp‘𝐷)
mapdrval.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
mapdrval.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdrval.r	⊢ (𝜑 → 𝑅 ∈ 𝑇)
mapdrval.e	⊢ (𝜑 → 𝑅 ⊆ 𝐶)
mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ))

**Assertion**
Ref	Expression
mapdrval	⊢ (𝜑 → (𝑀‘𝑄) = 𝑅)

Distinct variable groups: 𝑔,𝐹 𝑔,ℎ,𝐿 𝑔,𝑂,ℎ 𝑅,ℎ 𝑈,𝑔 Allowed substitution hints: 𝜑(𝑔,ℎ) 𝐶(𝑔,ℎ) 𝐷(𝑔,ℎ) 𝑄(𝑔,ℎ) 𝑅(𝑔) 𝑆(𝑔,ℎ) 𝑇(𝑔,ℎ) 𝑈(ℎ) 𝐹(ℎ) 𝐻(𝑔,ℎ) 𝐾(𝑔,ℎ) 𝑀(𝑔,ℎ) 𝑊(𝑔,ℎ)

Proof of Theorem mapdrval Dummy variables 𝑓 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdrval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdrval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdrval.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
4		mapdrval.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
5		mapdrval.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
6		mapdrval.o	. . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
7		mapdrval.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		mapdrval.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		mapdrval.d	. . . 4 ⊢ 𝐷 = (LDual‘𝑈)
10		mapdrval.t	. . . 4 ⊢ 𝑇 = (LSubSp‘𝐷)
11		mapdrval.c	. . . 4 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
12		mapdrval.q	. . . 4 ⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ))
13		mapdrval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑇)
14		mapdrval.e	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝐶)
15	1, 6, 2, 3, 4, 5, 9, 10, 11, 12, 8, 13, 14	lcfr 41544	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
16	1, 2, 3, 4, 5, 6, 7, 8, 15, 11	mapdvalc 41588	. 2 ⊢ (𝜑 → (𝑀‘𝑄) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄})
17		2fveq3 6927	. . . . 5 ⊢ (ℎ = 𝑖 → (𝑂‘(𝐿‘ℎ)) = (𝑂‘(𝐿‘𝑖)))
18	17	cbviunv 5063	. . . 4 ⊢ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)) = ∪ 𝑖 ∈ 𝑅 (𝑂‘(𝐿‘𝑖))
19	12, 18	eqtri 2768	. . 3 ⊢ 𝑄 = ∪ 𝑖 ∈ 𝑅 (𝑂‘(𝐿‘𝑖))
20		eqid 2740	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
21		eqid 2740	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
22		eqid 2740	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
23		eqid 2740	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
24		eqid 2740	. . 3 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
25	1, 6, 7, 2, 3, 4, 5, 9, 10, 11, 8, 13, 14, 19, 20, 21, 22, 23, 24	mapdrvallem3 41605	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)
26	16, 25	eqtrd 2780	1 ⊢ (𝜑 → (𝑀‘𝑄) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 ⊆ wss 3976 ∪ ciun 5015 ‘cfv 6575 Basecbs 17260 0_gc0g 17501 LSubSpclss 20954 LSpanclspn 20994 LSAtomsclsa 38932 LFnlclfn 39015 LKerclk 39043 LDualcld 39081 HLchlt 39308 LHypclh 39943 DVecHcdvh 41037 ocHcoch 41306 mapdcmpd 41583

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-riotaBAD 38911

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-undef 8316 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-0g 17503 df-mre 17646 df-mrc 17647 df-acs 17649 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cntz 19359 df-oppg 19388 df-lsm 19680 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20362 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20541 df-rlreg 20718 df-domn 20719 df-drng 20755 df-lmod 20884 df-lss 20955 df-lsp 20995 df-lvec 21127 df-lsatoms 38934 df-lshyp 38935 df-lcv 38977 df-lfl 39016 df-lkr 39044 df-ldual 39082 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-llines 39457 df-lplanes 39458 df-lvols 39459 df-lines 39460 df-psubsp 39462 df-pmap 39463 df-padd 39755 df-lhyp 39947 df-laut 39948 df-ldil 40063 df-ltrn 40064 df-trl 40118 df-tgrp 40702 df-tendo 40714 df-edring 40716 df-dveca 40962 df-disoa 40988 df-dvech 41038 df-dib 41098 df-dic 41132 df-dih 41188 df-doch 41307 df-djh 41354 df-mapd 41584

This theorem is referenced by: mapd1o 41607

W3C validator