lcdlss - Mathbox for Norm Megill

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

Theorem lcdlss 38759

Description: Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)

**Hypotheses**
Ref	Expression
lcdlss.h	⊢ 𝐻 = (LHyp‘𝐾)
lcdlss.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
lcdlss.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
lcdlss.s	⊢ 𝑆 = (LSubSp‘𝐶)
lcdlss.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcdlss.f	⊢ 𝐹 = (LFnl‘𝑈)
lcdlss.l	⊢ 𝐿 = (LKer‘𝑈)
lcdlss.d	⊢ 𝐷 = (LDual‘𝑈)
lcdlss.t	⊢ 𝑇 = (LSubSp‘𝐷)
lcdlss.b	⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcdlss.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
lcdlss	⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵))

Distinct variable groups: 𝐷,𝑓 𝑓,𝐹 𝑓,𝐾 𝑓,𝐿 𝑓,𝑂 𝑈,𝑓 𝑓,𝑊 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐶(𝑓) 𝑆(𝑓) 𝑇(𝑓) 𝐻(𝑓)

Proof of Theorem lcdlss Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcdlss.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝐶)
2		lcdlss.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
3		lcdlss.o	. . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
4		lcdlss.c	. . . . . . . 8 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
5		lcdlss.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lcdlss.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑈)
7		lcdlss.l	. . . . . . . 8 ⊢ 𝐿 = (LKer‘𝑈)
8		lcdlss.d	. . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈)
9		lcdlss.k	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		lcdlss.b	. . . . . . . 8 ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
11	2, 3, 4, 5, 6, 7, 8, 9, 10	lcdval2 38730	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐷 ↾_s 𝐵))
12	11	fveq2d 6677	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) = (LSubSp‘(𝐷 ↾_s 𝐵)))
13	1, 12	syl5eq 2871	. . . . 5 ⊢ (𝜑 → 𝑆 = (LSubSp‘(𝐷 ↾_s 𝐵)))
14	13	eleq2d 2901	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵))))
15	2, 5, 9	dvhlmod 38250	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
16	8, 15	lduallmod 36293	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ LMod)
17		lcdlss.t	. . . . . 6 ⊢ 𝑇 = (LSubSp‘𝐷)
18	2, 5, 3, 6, 7, 8, 17, 10, 9	lclkr 38673	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
19		eqid 2824	. . . . . 6 ⊢ (𝐷 ↾_s 𝐵) = (𝐷 ↾_s 𝐵)
20		eqid 2824	. . . . . 6 ⊢ (LSubSp‘(𝐷 ↾_s 𝐵)) = (LSubSp‘(𝐷 ↾_s 𝐵))
21	19, 17, 20	lsslss 19736	. . . . 5 ⊢ ((𝐷 ∈ LMod ∧ 𝐵 ∈ 𝑇) → (𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
22	16, 18, 21	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
23	14, 22	bitrd 281	. . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
24		elin 4172	. . . 4 ⊢ (𝑢 ∈ (𝑇 ∩ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵))
25		velpw 4547	. . . . 5 ⊢ (𝑢 ∈ 𝒫 𝐵 ↔ 𝑢 ⊆ 𝐵)
26	25	anbi2i 624	. . . 4 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))
27	24, 26	bitr2i 278	. . 3 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵) ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵))
28	23, 27	syl6bb 289	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵)))
29	28	eqrdv 2822	1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3145 ∩ cin 3938 ⊆ wss 3939 𝒫 cpw 4542 ‘cfv 6358 (class class class)co 7159 ↾_s cress 16487 LModclmod 19637 LSubSpclss 19706 LFnlclfn 36197 LKerclk 36225 LDualcld 36263 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 ocHcoch 38487 LCDualclcd 38726

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 df-lfl 36198 df-lkr 36226 df-ldual 36264 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tgrp 37883 df-tendo 37895 df-edring 37897 df-dveca 38143 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 df-doch 38488 df-djh 38535 df-lcdual 38727

This theorem is referenced by: lcdlss2N 38760 mapdrn2 38791

W3C validator