lcdlss - Mathbox for Norm Megill

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Theorem lcdlss 42118

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 42089	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐷 ↾_s 𝐵))
12	11	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) = (LSubSp‘(𝐷 ↾_s 𝐵)))
13	1, 12	eqtrid 2787	. . . . 5 ⊢ (𝜑 → 𝑆 = (LSubSp‘(𝐷 ↾_s 𝐵)))
14	13	eleq2d 2826	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵))))
15	2, 5, 9	dvhlmod 41609	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
16	8, 15	lduallmod 39652	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ LMod)
17		lcdlss.t	. . . . . 6 ⊢ 𝑇 = (LSubSp‘𝐷)
18	2, 5, 3, 6, 7, 8, 17, 10, 9	lclkr 42032	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
19		eqid 2740	. . . . . 6 ⊢ (𝐷 ↾_s 𝐵) = (𝐷 ↾_s 𝐵)
20		eqid 2740	. . . . . 6 ⊢ (LSubSp‘(𝐷 ↾_s 𝐵)) = (LSubSp‘(𝐷 ↾_s 𝐵))
21	19, 17, 20	lsslss 20958	. . . . 5 ⊢ ((𝐷 ∈ LMod ∧ 𝐵 ∈ 𝑇) → (𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
22	16, 18, 21	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝑢 ∈ (LSubSp‘(𝐷 ↾_s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
23	14, 22	bitrd 280	. . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)))
24		elin 3906	. . . 4 ⊢ (𝑢 ∈ (𝑇 ∩ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵))
25		velpw 4541	. . . . 5 ⊢ (𝑢 ∈ 𝒫 𝐵 ↔ 𝑢 ⊆ 𝐵)
26	25	anbi2i 629	. . . 4 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))
27	24, 26	bitr2i 277	. . 3 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵) ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵))
28	23, 27	bitrdi 288	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵)))
29	28	eqrdv 2738	1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 {crab 3392 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4536 ‘cfv 6492 (class class class)co 7363 ↾_s cress 17198 LModclmod 20857 LSubSpclss 20928 LFnlclfn 39556 LKerclk 39584 LDualcld 39622 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 ocHcoch 41846 LCDualclcd 42085

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-mre 17546 df-mrc 17547 df-acs 17549 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-oppg 19319 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20492 df-rlreg 20673 df-domn 20674 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lshyp 39476 df-lcv 39518 df-lfl 39557 df-lkr 39585 df-ldual 39623 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894 df-lcdual 42086

This theorem is referenced by: lcdlss2N 42119 mapdrn2 42150

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