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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkr | Structured version Visualization version GIF version |
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 fM 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.) |
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)) |
Ref | Expression |
---|---|
lclkr | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3650 | . . . 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 2610 | . . . 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 35417 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | 5, 6, 7, 11 | ldualvbase 33431 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
13 | 2, 4, 12 | 3sstr4d 3611 | . 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷)) |
14 | eqid 2610 | . . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
15 | eqid 2610 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
16 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
17 | 14, 15, 16, 5 | lfl0f 33374 | . . . . 5 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹) |
18 | 11, 17 | syl 17 | . . . 4 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹) |
19 | lclkr.o | . . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
20 | 8, 9, 19, 16, 10 | dochoc1 35668 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
21 | eqid 2610 | . . . . . . . 8 ⊢ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) | |
22 | lclkr.l | . . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈) | |
23 | 14, 15, 16, 5, 22 | lkr0f 33399 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹) → ((𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
24 | 11, 18, 23 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
25 | 21, 24 | mpbiri 247 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈)) |
26 | 25 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈))) |
27 | 26 | fveq2d 6107 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
28 | 20, 27, 25 | 3eqtr4d 2654 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
29 | 3 | lcfl1lem 35798 | . . . 4 ⊢ (((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶 ↔ (((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) |
30 | 18, 28, 29 | sylanbrc 695 | . . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶) |
31 | ne0i 3880 | . . 3 ⊢ (((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶 → 𝐶 ≠ ∅) | |
32 | 30, 31 | syl 17 | . 2 ⊢ (𝜑 → 𝐶 ≠ ∅) |
33 | eqid 2610 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
34 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
35 | eqid 2610 | . . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
36 | eqid 2610 | . . . . 5 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
37 | simpr1 1060 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷))) | |
38 | eqid 2610 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
39 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
40 | 14, 35, 6, 38, 39, 11 | ldualsbase 33438 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
41 | 40 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
42 | 37, 41 | eleqtrd 2690 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑈))) |
43 | simpr2 1061 | . . . . 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 35813 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐷)𝑎) ∈ 𝐶) |
45 | simpr3 1062 | . . . 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 35839 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶) |
47 | 46 | ralrimivvva 2955 | . 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 18756 | . 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 1239 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ⊆ wss 3540 ∅c0 3874 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LModclmod 18686 LSubSpclss 18753 LFnlclfn 33362 LKerclk 33390 LDualcld 33428 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 ocHcoch 35654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 |
This theorem is referenced by: lcdlvec 35898 lcd0v 35918 lcdlss 35926 lcdlsp 35928 mapdunirnN 35957 |
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