lclkr - Mathbox for Norm Megill

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Theorem lclkr 41516

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 4090	. . . 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 2735	. . . 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 41093	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	5, 6, 7, 11	ldualvbase 39108	. . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
13	2, 4, 12	3sstr4d 4043	. 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷))
14		eqid 2735	. . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
15		eqid 2735	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
16		eqid 2735	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
17	14, 15, 16, 5	lfl0f 39051	. . . . 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 41344	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		eqid 2735	. . . . . . . 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 39076	. . . . . . . . 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, 17, 23	syl2anc2 585	. . . . . . . 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 258	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈))
26	25	fveq2d 6911	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈)))
27	26	fveq2d 6911	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
28	20, 27, 25	3eqtr4d 2785	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
29	3	lcfl1lem 41474	. . . 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 583	. . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶)
31	30	ne0d 4348	. 2 ⊢ (𝜑 → 𝐶 ≠ ∅)
32		eqid 2735	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
33	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
34		eqid 2735	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
35		eqid 2735	. . . . 5 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
36		simpr1 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷)))
37		eqid 2735	. . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
38		eqid 2735	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
39	14, 34, 6, 37, 38, 11	ldualsbase 39115	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
41	36, 40	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑈)))
42		simpr2 1194	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶)
43	8, 19, 9, 5, 22, 6, 14, 34, 35, 3, 33, 41, 42	lclkrlem1 41489	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑥( ·_𝑠 ‘𝐷)𝑎) ∈ 𝐶)
44		simpr3 1195	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶)
45	8, 19, 9, 5, 22, 6, 32, 3, 33, 43, 44	lclkrlem2 41515	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
46	45	ralrimivvva 3203	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
47		lclkr.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝐷)
48	37, 38, 7, 32, 35, 47	islss 20950	. 2 ⊢ (𝐶 ∈ 𝑆 ↔ (𝐶 ⊆ (Base‘𝐷) ∧ 𝐶 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶))
49	13, 31, 46, 48	syl3anbrc 1342	1 ⊢ (𝜑 → 𝐶 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 ⊆ wss 3963 ∅c0 4339 {csn 4631 × cxp 5687 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 Scalarcsca 17301 ·_𝑠 cvsca 17302 0_gc0g 17486 LModclmod 20875 LSubSpclss 20947 LFnlclfn 39039 LKerclk 39067 LDualcld 39105 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 ocHcoch 41330

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

This theorem is referenced by: lcdlvec 41574 lcd0v 41594 lcdlss 41602 lcdlsp 41604 mapdunirnN 41633

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