mapdrval - Mathbox for Norm Megill

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Theorem mapdrval 41633

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 41571	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
16	1, 2, 3, 4, 5, 6, 7, 8, 15, 11	mapdvalc 41615	. 2 ⊢ (𝜑 → (𝑀‘𝑄) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄})
17		2fveq3 6870	. . . . 5 ⊢ (ℎ = 𝑖 → (𝑂‘(𝐿‘ℎ)) = (𝑂‘(𝐿‘𝑖)))
18	17	cbviunv 5012	. . . 4 ⊢ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)) = ∪ 𝑖 ∈ 𝑅 (𝑂‘(𝐿‘𝑖))
19	12, 18	eqtri 2753	. . 3 ⊢ 𝑄 = ∪ 𝑖 ∈ 𝑅 (𝑂‘(𝐿‘𝑖))
20		eqid 2730	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
21		eqid 2730	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
22		eqid 2730	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
23		eqid 2730	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
24		eqid 2730	. . 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 41632	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)
26	16, 25	eqtrd 2765	1 ⊢ (𝜑 → (𝑀‘𝑄) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3411 ⊆ wss 3922 ∪ ciun 4963 ‘cfv 6519 Basecbs 17185 0_gc0g 17408 LSubSpclss 20843 LSpanclspn 20883 LSAtomsclsa 38959 LFnlclfn 39042 LKerclk 39070 LDualcld 39108 HLchlt 39335 LHypclh 39970 DVecHcdvh 41064 ocHcoch 41333 mapdcmpd 41610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163 ax-riotaBAD 38938

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-int 4919 df-iun 4965 df-iin 4966 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7660 df-om 7851 df-1st 7977 df-2nd 7978 df-tpos 8214 df-undef 8261 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-2o 8444 df-er 8682 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-n0 12459 df-z 12546 df-uz 12810 df-fz 13482 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17410 df-mre 17553 df-mrc 17554 df-acs 17556 df-proset 18261 df-poset 18280 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-subg 19061 df-cntz 19255 df-oppg 19284 df-lsm 19572 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-nzr 20428 df-rlreg 20609 df-domn 20610 df-drng 20646 df-lmod 20774 df-lss 20844 df-lsp 20884 df-lvec 21016 df-lsatoms 38961 df-lshyp 38962 df-lcv 39004 df-lfl 39043 df-lkr 39071 df-ldual 39109 df-oposet 39161 df-ol 39163 df-oml 39164 df-covers 39251 df-ats 39252 df-atl 39283 df-cvlat 39307 df-hlat 39336 df-llines 39484 df-lplanes 39485 df-lvols 39486 df-lines 39487 df-psubsp 39489 df-pmap 39490 df-padd 39782 df-lhyp 39974 df-laut 39975 df-ldil 40090 df-ltrn 40091 df-trl 40145 df-tgrp 40729 df-tendo 40741 df-edring 40743 df-dveca 40989 df-disoa 41015 df-dvech 41065 df-dib 41125 df-dic 41159 df-dih 41215 df-doch 41334 df-djh 41381 df-mapd 41611

This theorem is referenced by: mapd1o 41634

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