lcdlss - Mathbox for Norm Megill

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Theorem lcdlss 41602

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 41573	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐷 ↾_s 𝐵))
12	11	fveq2d 6911	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) = (LSubSp‘(𝐷 ↾_s 𝐵)))
13	1, 12	eqtrid 2787	. . . . 5 ⊢ (𝜑 → 𝑆 = (LSubSp‘(𝐷 ↾_s 𝐵)))
14	13	eleq2d 2825	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵))))
15	2, 5, 9	dvhlmod 41093	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
16	8, 15	lduallmod 39135	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ LMod)
17		lcdlss.t	. . . . . 6 ⊢ 𝑇 = (LSubSp‘𝐷)
18	2, 5, 3, 6, 7, 8, 17, 10, 9	lclkr 41516	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
19		eqid 2735	. . . . . 6 ⊢ (𝐷 ↾_s 𝐵) = (𝐷 ↾_s 𝐵)
20		eqid 2735	. . . . . 6 ⊢ (LSubSp‘(𝐷 ↾_s 𝐵)) = (LSubSp‘(𝐷 ↾_s 𝐵))
21	19, 17, 20	lsslss 20977	. . . . 5 ⊢ ((𝐷 ∈ LMod ∧ 𝐵 ∈ 𝑇) → (𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
22	16, 18, 21	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
23	14, 22	bitrd 279	. . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
24		elin 3979	. . . 4 ⊢ (𝑢 ∈ (𝑇 ∩ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵))
25		velpw 4610	. . . . 5 ⊢ (𝑢 ∈ 𝒫 𝐵 ↔ 𝑢 ⊆ 𝐵)
26	25	anbi2i 623	. . . 4 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))
27	24, 26	bitr2i 276	. . 3 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵) ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵))
28	23, 27	bitrdi 287	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵)))
29	28	eqrdv 2733	1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ‘cfv 6563 (class class class)co 7431 ↾_s cress 17274 LModclmod 20875 LSubSpclss 20947 LFnlclfn 39039 LKerclk 39067 LDualcld 39105 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 ocHcoch 41330 LCDualclcd 41569

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570

This theorem is referenced by: lcdlss2N 41603 mapdrn2 41634

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