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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 36249 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍))) |
19 | riotacl 7134 | . . . 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 2828 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | |
23 | 22 | rexbidv 3300 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
24 | 23 | riotabidv 7119 | . . . . 5 ⊢ (𝑥 = 𝑎 → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
25 | 24 | cbvmptv 5172 | . . . 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 2847 | . . 3 ⊢ 𝐺 = (𝑎 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
27 | 20, 26 | fmptd 6881 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
28 | eqid 2824 | . . . 4 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
29 | 1, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 28, 21 | lshpkrlem6 36255 | . . 3 ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) |
30 | 29 | ralrimivvva 3195 | . 2 ⊢ (𝜑 → ∀𝑙 ∈ 𝐾 ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ 𝑉 (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) |
31 | eqid 2824 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
32 | eqid 2824 | . . . 4 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
33 | lshpkr.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
34 | 1, 2, 15, 17, 16, 31, 32, 33 | islfl 36200 | . . 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 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 ∃!wreu 3143 {csn 4570 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 ℩crio 7116 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 Scalarcsca 16571 ·𝑠 cvsca 16572 0gc0g 16716 LSSumclsm 18762 LSpanclspn 19746 LVecclvec 19877 LSHypclsh 36115 LFnlclfn 36197 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lshyp 36117 df-lfl 36198 |
This theorem is referenced by: lshpkr 36257 lshpkrex 36258 dochflcl 38615 |
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