| Mathbox for Norm Megill |
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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 4080 | . . . 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 2737 | . . . 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 41112 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 12 | 5, 6, 7, 11 | ldualvbase 39127 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
| 13 | 2, 4, 12 | 3sstr4d 4039 | . 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷)) |
| 14 | eqid 2737 | . . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
| 15 | eqid 2737 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
| 16 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
| 17 | 14, 15, 16, 5 | lfl0f 39070 | . . . . 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 41363 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
| 21 | eqid 2737 | . . . . . . . 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 39095 | . . . . . . . . 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, 17, 23 | syl2anc2 585 | . . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
| 25 | 21, 24 | mpbiri 258 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈)) |
| 26 | 25 | fveq2d 6910 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈))) |
| 27 | 26 | fveq2d 6910 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
| 28 | 20, 27, 25 | 3eqtr4d 2787 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
| 29 | 3 | lcfl1lem 41493 | . . . 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 583 | . . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶) |
| 31 | 30 | ne0d 4342 | . 2 ⊢ (𝜑 → 𝐶 ≠ ∅) |
| 32 | eqid 2737 | . . . 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 2737 | . . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
| 35 | eqid 2737 | . . . . 5 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
| 36 | simpr1 1195 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷))) | |
| 37 | eqid 2737 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
| 38 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
| 39 | 14, 34, 6, 37, 38, 11 | ldualsbase 39134 | . . . . . . 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 2843 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑈))) |
| 42 | simpr2 1196 | . . . . 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 41508 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐷)𝑎) ∈ 𝐶) |
| 44 | simpr3 1197 | . . . 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 41534 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶) |
| 46 | 45 | ralrimivvva 3205 | . 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 20932 | . 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 1344 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 ⊆ wss 3951 ∅c0 4333 {csn 4626 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LModclmod 20858 LSubSpclss 20929 LFnlclfn 39058 LKerclk 39086 LDualcld 39124 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 ocHcoch 41349 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-nzr 20513 df-rlreg 20694 df-domn 20695 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lshyp 38978 df-lcv 39020 df-lfl 39059 df-lkr 39087 df-ldual 39125 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 |
| This theorem is referenced by: lcdlvec 41593 lcd0v 41613 lcdlss 41621 lcdlsp 41623 mapdunirnN 41652 |
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