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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpkrcl | Structured version Visualization version GIF version | ||
| Description: The set 𝐺 defined by hyperplane 𝑈 is a linear functional. (Contributed by NM, 17-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| lshpkr.v | ⊢ 𝑉 = (Base‘𝑊) | 
| lshpkr.a | ⊢ + = (+g‘𝑊) | 
| lshpkr.n | ⊢ 𝑁 = (LSpan‘𝑊) | 
| lshpkr.p | ⊢ ⊕ = (LSSum‘𝑊) | 
| lshpkr.h | ⊢ 𝐻 = (LSHyp‘𝑊) | 
| lshpkr.w | ⊢ (𝜑 → 𝑊 ∈ LVec) | 
| lshpkr.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) | 
| lshpkr.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) | 
| lshpkr.e | ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | 
| lshpkr.d | ⊢ 𝐷 = (Scalar‘𝑊) | 
| lshpkr.k | ⊢ 𝐾 = (Base‘𝐷) | 
| lshpkr.t | ⊢ · = ( ·𝑠 ‘𝑊) | 
| lshpkr.g | ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) | 
| lshpkr.f | ⊢ 𝐹 = (LFnl‘𝑊) | 
| Ref | Expression | 
|---|---|
| lshpkrcl | ⊢ (𝜑 → 𝐺 ∈ 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lshpkr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 2 | lshpkr.a | . . . . 5 ⊢ + = (+g‘𝑊) | |
| 3 | lshpkr.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 4 | lshpkr.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝑊) | |
| 5 | lshpkr.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
| 6 | lshpkr.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
| 7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑊 ∈ LVec) | 
| 8 | lshpkr.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
| 9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑈 ∈ 𝐻) | 
| 10 | lshpkr.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
| 11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑍 ∈ 𝑉) | 
| 12 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
| 13 | lshpkr.e | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | |
| 14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | 
| 15 | lshpkr.d | . . . . 5 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 16 | lshpkr.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐷) | |
| 17 | lshpkr.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 18 | 1, 2, 3, 4, 5, 7, 9, 11, 12, 14, 15, 16, 17 | lshpsmreu 39111 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍))) | 
| 19 | riotacl 7406 | . . . 4 ⊢ (∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)) → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍))) ∈ 𝐾) | |
| 20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍))) ∈ 𝐾) | 
| 21 | lshpkr.g | . . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) | |
| 22 | eqeq1 2740 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | |
| 23 | 22 | rexbidv 3178 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | 
| 24 | 23 | riotabidv 7391 | . . . . 5 ⊢ (𝑥 = 𝑎 → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | 
| 25 | 24 | cbvmptv 5254 | . . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) = (𝑎 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | 
| 26 | 21, 25 | eqtri 2764 | . . 3 ⊢ 𝐺 = (𝑎 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | 
| 27 | 20, 26 | fmptd 7133 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) | 
| 28 | eqid 2736 | . . . 4 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
| 29 | 1, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 28, 21 | lshpkrlem6 39117 | . . 3 ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) | 
| 30 | 29 | ralrimivvva 3204 | . 2 ⊢ (𝜑 → ∀𝑙 ∈ 𝐾 ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ 𝑉 (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) | 
| 31 | eqid 2736 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
| 32 | eqid 2736 | . . . 4 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
| 33 | lshpkr.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 34 | 1, 2, 15, 17, 16, 31, 32, 33 | islfl 39062 | . . 3 ⊢ (𝑊 ∈ LVec → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑙 ∈ 𝐾 ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ 𝑉 (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))))) | 
| 35 | 6, 34 | syl 17 | . 2 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑙 ∈ 𝐾 ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ 𝑉 (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))))) | 
| 36 | 27, 30, 35 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ∃!wreu 3377 {csn 4625 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17485 LSSumclsm 19653 LSpanclspn 20970 LVecclvec 21102 LSHypclsh 38977 LFnlclfn 39059 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cntz 19336 df-lsm 19655 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 df-lshyp 38979 df-lfl 39060 | 
| This theorem is referenced by: lshpkr 39119 lshpkrex 39120 dochflcl 41478 | 
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