lclkr - Mathbox for Norm Megill

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Theorem lclkr 36822

Description: The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of [Holland95] p. 218, line 20, stating "The f_M that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015.)

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

**Assertion**
Ref	Expression
lclkr	⊢ (𝜑 → 𝐶 ∈ 𝑆)

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

Proof of Theorem lclkr Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . 4 ⊢ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ⊆ 𝐹
2	1	a1i 11	. . 3 ⊢ (𝜑 → {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ⊆ 𝐹)
3		lclkr.c	. . . 4 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
5		lclkr.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑈)
6		lclkr.d	. . . 4 ⊢ 𝐷 = (LDual‘𝑈)
7		eqid 2622	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		lclkr.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
9		lclkr.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		lclkr.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	8, 9, 10	dvhlmod 36399	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	5, 6, 7, 11	ldualvbase 34413	. . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
13	2, 4, 12	3sstr4d 3648	. 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷))
14		eqid 2622	. . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
15		eqid 2622	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
16		eqid 2622	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
17	14, 15, 16, 5	lfl0f 34356	. . . . 5 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹)
18	11, 17	syl 17	. . . 4 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹)
19		lclkr.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
20	8, 9, 19, 16, 10	dochoc1 36650	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		eqid 2622	. . . . . . . 8 ⊢ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})
22		lclkr.l	. . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈)
23	14, 15, 16, 5, 22	lkr0f 34381	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹) → ((𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
24	11, 18, 23	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
25	21, 24	mpbiri 248	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈))
26	25	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈)))
27	26	fveq2d 6195	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
28	20, 27, 25	3eqtr4d 2666	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
29	3	lcfl1lem 36780	. . . 4 ⊢ (((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶 ↔ (((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))))
30	18, 28, 29	sylanbrc 698	. . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶)
31		ne0i 3921	. . 3 ⊢ (((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶 → 𝐶 ≠ ∅)
32	30, 31	syl 17	. 2 ⊢ (𝜑 → 𝐶 ≠ ∅)
33		eqid 2622	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
34	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35		eqid 2622	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
36		eqid 2622	. . . . 5 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
37		simpr1 1067	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷)))
38		eqid 2622	. . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
39		eqid 2622	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
40	14, 35, 6, 38, 39, 11	ldualsbase 34420	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
41	40	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
42	37, 41	eleqtrd 2703	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑈)))
43		simpr2 1068	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶)
44	8, 19, 9, 5, 22, 6, 14, 35, 36, 3, 34, 42, 43	lclkrlem1 36795	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑥( ·_𝑠 ‘𝐷)𝑎) ∈ 𝐶)
45		simpr3 1069	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶)
46	8, 19, 9, 5, 22, 6, 33, 3, 34, 44, 45	lclkrlem2 36821	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
47	46	ralrimivvva 2972	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
48		lclkr.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝐷)
49	38, 39, 7, 33, 36, 48	islss 18935	. 2 ⊢ (𝐶 ∈ 𝑆 ↔ (𝐶 ⊆ (Base‘𝐷) ∧ 𝐶 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶))
50	13, 32, 47, 49	syl3anbrc 1246	1 ⊢ (𝜑 → 𝐶 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ⊆ wss 3574 ∅c0 3915 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 LModclmod 18863 LSubSpclss 18932 LFnlclfn 34344 LKerclk 34372 LDualcld 34410 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 ocHcoch 36636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684

This theorem is referenced by: lcdlvec 36880 lcd0v 36900 lcdlss 36908 lcdlsp 36910 mapdunirnN 36939

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