mapdrval - Mathbox for Norm Megill

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Theorem mapdrval 41583

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 41521	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
16	1, 2, 3, 4, 5, 6, 7, 8, 15, 11	mapdvalc 41565	. 2 ⊢ (𝜑 → (𝑀‘𝑄) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄})
17		2fveq3 6890	. . . . 5 ⊢ (ℎ = 𝑖 → (𝑂‘(𝐿‘ℎ)) = (𝑂‘(𝐿‘𝑖)))
18	17	cbviunv 5020	. . . 4 ⊢ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)) = ∪ 𝑖 ∈ 𝑅 (𝑂‘(𝐿‘𝑖))
19	12, 18	eqtri 2757	. . 3 ⊢ 𝑄 = ∪ 𝑖 ∈ 𝑅 (𝑂‘(𝐿‘𝑖))
20		eqid 2734	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
21		eqid 2734	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
22		eqid 2734	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
23		eqid 2734	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
24		eqid 2734	. . 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 41582	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)
26	16, 25	eqtrd 2769	1 ⊢ (𝜑 → (𝑀‘𝑄) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3419 ⊆ wss 3931 ∪ ciun 4971 ‘cfv 6540 Basecbs 17228 0_gc0g 17454 LSubSpclss 20896 LSpanclspn 20936 LSAtomsclsa 38909 LFnlclfn 38992 LKerclk 39020 LDualcld 39058 HLchlt 39285 LHypclh 39920 DVecHcdvh 41014 ocHcoch 41283 mapdcmpd 41560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38888

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6493 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7678 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11475 df-neg 11476 df-nn 12248 df-2 12310 df-3 12311 df-4 12312 df-5 12313 df-6 12314 df-n0 12509 df-z 12596 df-uz 12860 df-fz 13529 df-struct 17165 df-sets 17182 df-slot 17200 df-ndx 17212 df-base 17229 df-ress 17252 df-plusg 17285 df-mulr 17286 df-sca 17288 df-vsca 17289 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18445 df-clat 18512 df-mgm 18621 df-sgrp 18700 df-mnd 18716 df-submnd 18765 df-grp 18922 df-minusg 18923 df-sbg 18924 df-subg 19109 df-cntz 19303 df-oppg 19332 df-lsm 19621 df-cmn 19767 df-abl 19768 df-mgp 20105 df-rng 20117 df-ur 20146 df-ring 20199 df-oppr 20301 df-dvdsr 20324 df-unit 20325 df-invr 20355 df-dvr 20368 df-nzr 20480 df-rlreg 20661 df-domn 20662 df-drng 20698 df-lmod 20827 df-lss 20897 df-lsp 20937 df-lvec 21069 df-lsatoms 38911 df-lshyp 38912 df-lcv 38954 df-lfl 38993 df-lkr 39021 df-ldual 39059 df-oposet 39111 df-ol 39113 df-oml 39114 df-covers 39201 df-ats 39202 df-atl 39233 df-cvlat 39257 df-hlat 39286 df-llines 39434 df-lplanes 39435 df-lvols 39436 df-lines 39437 df-psubsp 39439 df-pmap 39440 df-padd 39732 df-lhyp 39924 df-laut 39925 df-ldil 40040 df-ltrn 40041 df-trl 40095 df-tgrp 40679 df-tendo 40691 df-edring 40693 df-dveca 40939 df-disoa 40965 df-dvech 41015 df-dib 41075 df-dic 41109 df-dih 41165 df-doch 41284 df-djh 41331 df-mapd 41561

This theorem is referenced by: mapd1o 41584

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