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Mathbox for Norm Megill |
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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 39091 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍))) |
19 | riotacl 7405 | . . . 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 2739 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑎 = (𝑦 + (𝑘 · 𝑍)))) | |
23 | 22 | rexbidv 3177 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
24 | 23 | riotabidv 7390 | . . . . 5 ⊢ (𝑥 = 𝑎 → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
25 | 24 | cbvmptv 5261 | . . . 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 2763 | . . 3 ⊢ 𝐺 = (𝑎 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑎 = (𝑦 + (𝑘 · 𝑍)))) |
27 | 20, 26 | fmptd 7134 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
28 | eqid 2735 | . . . 4 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
29 | 1, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 28, 21 | lshpkrlem6 39097 | . . 3 ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) |
30 | 29 | ralrimivvva 3203 | . 2 ⊢ (𝜑 → ∀𝑙 ∈ 𝐾 ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ 𝑉 (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) |
31 | eqid 2735 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
32 | eqid 2735 | . . . 4 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
33 | lshpkr.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
34 | 1, 2, 15, 17, 16, 31, 32, 33 | islfl 39042 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 {csn 4631 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LSSumclsm 19667 LSpanclspn 20987 LVecclvec 21119 LSHypclsh 38957 LFnlclfn 39039 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lshyp 38959 df-lfl 39040 |
This theorem is referenced by: lshpkr 39099 lshpkrex 39100 dochflcl 41458 |
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