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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 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑊 ∈ LVec) |
8 | lshpkr.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑈 ∈ 𝐻) |
10 | lshpkr.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑍 ∈ 𝑉) |
12 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
13 | lshpkr.e | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | |
14 | 13 | adantr 483 | . . . . 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 36239 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍))) |
19 | riotacl 7125 | . . . 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 2825 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | |
23 | 22 | rexbidv 3297 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
24 | 23 | riotabidv 7110 | . . . . 5 ⊢ (𝑥 = 𝑎 → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
25 | 24 | cbvmptv 5161 | . . . 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 2844 | . . 3 ⊢ 𝐺 = (𝑎 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
27 | 20, 26 | fmptd 6872 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
28 | eqid 2821 | . . . 4 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
29 | 1, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 28, 21 | lshpkrlem6 36245 | . . 3 ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) |
30 | 29 | ralrimivvva 3192 | . 2 ⊢ (𝜑 → ∀𝑙 ∈ 𝐾 ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ 𝑉 (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) |
31 | eqid 2821 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
32 | eqid 2821 | . . . 4 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
33 | lshpkr.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
34 | 1, 2, 15, 17, 16, 31, 32, 33 | islfl 36190 | . . 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 711 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∃!wreu 3140 {csn 4560 ↦ cmpt 5138 ⟶wf 6345 ‘cfv 6349 ℩crio 7107 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 LSSumclsm 18753 LSpanclspn 19737 LVecclvec 19868 LSHypclsh 36105 LFnlclfn 36187 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lshyp 36107 df-lfl 36188 |
This theorem is referenced by: lshpkr 36247 lshpkrex 36248 dochflcl 38605 |
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