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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 πΊ defined by hyperplane π is a linear functional. (Contributed by NM, 17-Jul-2014.) |
Ref | Expression |
---|---|
lshpkr.v | β’ π = (Baseβπ) |
lshpkr.a | β’ + = (+gβπ) |
lshpkr.n | β’ π = (LSpanβπ) |
lshpkr.p | β’ β = (LSSumβπ) |
lshpkr.h | β’ π» = (LSHypβπ) |
lshpkr.w | β’ (π β π β LVec) |
lshpkr.u | β’ (π β π β π») |
lshpkr.z | β’ (π β π β π) |
lshpkr.e | β’ (π β (π β (πβ{π})) = π) |
lshpkr.d | β’ π· = (Scalarβπ) |
lshpkr.k | β’ πΎ = (Baseβπ·) |
lshpkr.t | β’ Β· = ( Β·π βπ) |
lshpkr.g | β’ πΊ = (π₯ β π β¦ (β©π β πΎ βπ¦ β π π₯ = (π¦ + (π Β· π)))) |
lshpkr.f | β’ πΉ = (LFnlβπ) |
Ref | Expression |
---|---|
lshpkrcl | β’ (π β πΊ β πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpkr.v | . . . . 5 β’ π = (Baseβπ) | |
2 | lshpkr.a | . . . . 5 β’ + = (+gβπ) | |
3 | lshpkr.n | . . . . 5 β’ π = (LSpanβπ) | |
4 | lshpkr.p | . . . . 5 β’ β = (LSSumβπ) | |
5 | lshpkr.h | . . . . 5 β’ π» = (LSHypβπ) | |
6 | lshpkr.w | . . . . . 6 β’ (π β π β LVec) | |
7 | 6 | adantr 479 | . . . . 5 β’ ((π β§ π β π) β π β LVec) |
8 | lshpkr.u | . . . . . 6 β’ (π β π β π») | |
9 | 8 | adantr 479 | . . . . 5 β’ ((π β§ π β π) β π β π») |
10 | lshpkr.z | . . . . . 6 β’ (π β π β π) | |
11 | 10 | adantr 479 | . . . . 5 β’ ((π β§ π β π) β π β π) |
12 | simpr 483 | . . . . 5 β’ ((π β§ π β π) β π β π) | |
13 | lshpkr.e | . . . . . 6 β’ (π β (π β (πβ{π})) = π) | |
14 | 13 | adantr 479 | . . . . 5 β’ ((π β§ π β π) β (π β (πβ{π})) = π) |
15 | lshpkr.d | . . . . 5 β’ π· = (Scalarβπ) | |
16 | lshpkr.k | . . . . 5 β’ πΎ = (Baseβπ·) | |
17 | lshpkr.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
18 | 1, 2, 3, 4, 5, 7, 9, 11, 12, 14, 15, 16, 17 | lshpsmreu 38613 | . . . 4 β’ ((π β§ π β π) β β!π β πΎ βπ¦ β π π = (π¦ + (π Β· π))) |
19 | riotacl 7400 | . . . 4 β’ (β!π β πΎ βπ¦ β π π = (π¦ + (π Β· π)) β (β©π β πΎ βπ¦ β π π = (π¦ + (π Β· π))) β πΎ) | |
20 | 18, 19 | syl 17 | . . 3 β’ ((π β§ π β π) β (β©π β πΎ βπ¦ β π π = (π¦ + (π Β· π))) β πΎ) |
21 | lshpkr.g | . . . 4 β’ πΊ = (π₯ β π β¦ (β©π β πΎ βπ¦ β π π₯ = (π¦ + (π Β· π)))) | |
22 | eqeq1 2732 | . . . . . . 7 β’ (π₯ = π β (π₯ = (π¦ + (π Β· π)) β π = (π¦ + (π Β· π)))) | |
23 | 22 | rexbidv 3176 | . . . . . 6 β’ (π₯ = π β (βπ¦ β π π₯ = (π¦ + (π Β· π)) β βπ¦ β π π = (π¦ + (π Β· π)))) |
24 | 23 | riotabidv 7384 | . . . . 5 β’ (π₯ = π β (β©π β πΎ βπ¦ β π π₯ = (π¦ + (π Β· π))) = (β©π β πΎ βπ¦ β π π = (π¦ + (π Β· π)))) |
25 | 24 | cbvmptv 5265 | . . . 4 β’ (π₯ β π β¦ (β©π β πΎ βπ¦ β π π₯ = (π¦ + (π Β· π)))) = (π β π β¦ (β©π β πΎ βπ¦ β π π = (π¦ + (π Β· π)))) |
26 | 21, 25 | eqtri 2756 | . . 3 β’ πΊ = (π β π β¦ (β©π β πΎ βπ¦ β π π = (π¦ + (π Β· π)))) |
27 | 20, 26 | fmptd 7129 | . 2 β’ (π β πΊ:πβΆπΎ) |
28 | eqid 2728 | . . . 4 β’ (0gβπ·) = (0gβπ·) | |
29 | 1, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 28, 21 | lshpkrlem6 38619 | . . 3 β’ ((π β§ (π β πΎ β§ π’ β π β§ π£ β π)) β (πΊβ((π Β· π’) + π£)) = ((π(.rβπ·)(πΊβπ’))(+gβπ·)(πΊβπ£))) |
30 | 29 | ralrimivvva 3201 | . 2 β’ (π β βπ β πΎ βπ’ β π βπ£ β π (πΊβ((π Β· π’) + π£)) = ((π(.rβπ·)(πΊβπ’))(+gβπ·)(πΊβπ£))) |
31 | eqid 2728 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
32 | eqid 2728 | . . . 4 β’ (.rβπ·) = (.rβπ·) | |
33 | lshpkr.f | . . . 4 β’ πΉ = (LFnlβπ) | |
34 | 1, 2, 15, 17, 16, 31, 32, 33 | islfl 38564 | . . 3 β’ (π β LVec β (πΊ β πΉ β (πΊ:πβΆπΎ β§ βπ β πΎ βπ’ β π βπ£ β π (πΊβ((π Β· π’) + π£)) = ((π(.rβπ·)(πΊβπ’))(+gβπ·)(πΊβπ£))))) |
35 | 6, 34 | syl 17 | . 2 β’ (π β (πΊ β πΉ β (πΊ:πβΆπΎ β§ βπ β πΎ βπ’ β π βπ£ β π (πΊβ((π Β· π’) + π£)) = ((π(.rβπ·)(πΊβπ’))(+gβπ·)(πΊβπ£))))) |
36 | 27, 30, 35 | mpbir2and 711 | 1 β’ (π β πΊ β πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 βwrex 3067 β!wreu 3372 {csn 4632 β¦ cmpt 5235 βΆwf 6549 βcfv 6553 β©crio 7381 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 Scalarcsca 17243 Β·π cvsca 17244 0gc0g 17428 LSSumclsm 19596 LSpanclspn 20862 LVecclvec 20994 LSHypclsh 38479 LFnlclfn 38561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lshyp 38481 df-lfl 38562 |
This theorem is referenced by: lshpkr 38621 lshpkrex 38622 dochflcl 40980 |
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