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 41608

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 41579	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐷 ↾_s 𝐵))
12	11	fveq2d 6864	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) = (LSubSp‘(𝐷 ↾_s 𝐵)))
13	1, 12	eqtrid 2777	. . . . 5 ⊢ (𝜑 → 𝑆 = (LSubSp‘(𝐷 ↾_s 𝐵)))
14	13	eleq2d 2815	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵))))
15	2, 5, 9	dvhlmod 41099	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
16	8, 15	lduallmod 39141	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ LMod)
17		lcdlss.t	. . . . . 6 ⊢ 𝑇 = (LSubSp‘𝐷)
18	2, 5, 3, 6, 7, 8, 17, 10, 9	lclkr 41522	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
19		eqid 2730	. . . . . 6 ⊢ (𝐷 ↾_s 𝐵) = (𝐷 ↾_s 𝐵)
20		eqid 2730	. . . . . 6 ⊢ (LSubSp‘(𝐷 ↾_s 𝐵)) = (LSubSp‘(𝐷 ↾_s 𝐵))
21	19, 17, 20	lsslss 20873	. . . . 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 3932	. . . 4 ⊢ (𝑢 ∈ (𝑇 ∩ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵))
25		velpw 4570	. . . . 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 2728	1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3408 ∩ cin 3915 ⊆ wss 3916 𝒫 cpw 4565 ‘cfv 6513 (class class class)co 7389 ↾_s cress 17206 LModclmod 20772 LSubSpclss 20843 LFnlclfn 39045 LKerclk 39073 LDualcld 39111 HLchlt 39338 LHypclh 39973 DVecHcdvh 41067 ocHcoch 41336 LCDualclcd 41575

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 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-riotaBAD 38941

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 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-n0 12449 df-z 12536 df-uz 12800 df-fz 13475 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 38964 df-lshyp 38965 df-lcv 39007 df-lfl 39046 df-lkr 39074 df-ldual 39112 df-oposet 39164 df-ol 39166 df-oml 39167 df-covers 39254 df-ats 39255 df-atl 39286 df-cvlat 39310 df-hlat 39339 df-llines 39487 df-lplanes 39488 df-lvols 39489 df-lines 39490 df-psubsp 39492 df-pmap 39493 df-padd 39785 df-lhyp 39977 df-laut 39978 df-ldil 40093 df-ltrn 40094 df-trl 40148 df-tgrp 40732 df-tendo 40744 df-edring 40746 df-dveca 40992 df-disoa 41018 df-dvech 41068 df-dib 41128 df-dic 41162 df-dih 41218 df-doch 41337 df-djh 41384 df-lcdual 41576

This theorem is referenced by: lcdlss2N 41609 mapdrn2 41640

W3C validator