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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 480 | . . . . 5 β’ ((π β§ π β π) β π β LVec) |
8 | lshpkr.u | . . . . . 6 β’ (π β π β π») | |
9 | 8 | adantr 480 | . . . . 5 β’ ((π β§ π β π) β π β π») |
10 | lshpkr.z | . . . . . 6 β’ (π β π β π) | |
11 | 10 | adantr 480 | . . . . 5 β’ ((π β§ π β π) β π β π) |
12 | simpr 484 | . . . . 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 480 | . . . . 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 38491 | . . . 4 β’ ((π β§ π β π) β β!π β πΎ βπ¦ β π π = (π¦ + (π Β· π))) |
19 | riotacl 7378 | . . . 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 2730 | . . . . . . 7 β’ (π₯ = π β (π₯ = (π¦ + (π Β· π)) β π = (π¦ + (π Β· π)))) | |
23 | 22 | rexbidv 3172 | . . . . . 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 7362 | . . . . 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 5254 | . . . 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 2754 | . . 3 β’ πΊ = (π β π β¦ (β©π β πΎ βπ¦ β π π = (π¦ + (π Β· π)))) |
27 | 20, 26 | fmptd 7108 | . 2 β’ (π β πΊ:πβΆπΎ) |
28 | eqid 2726 | . . . 4 β’ (0gβπ·) = (0gβπ·) | |
29 | 1, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 28, 21 | lshpkrlem6 38497 | . . 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 3197 | . 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 2726 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
32 | eqid 2726 | . . . 4 β’ (.rβπ·) = (.rβπ·) | |
33 | lshpkr.f | . . . 4 β’ πΉ = (LFnlβπ) | |
34 | 1, 2, 15, 17, 16, 31, 32, 33 | islfl 38442 | . . 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 710 | 1 β’ (π β πΊ β πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 β!wreu 3368 {csn 4623 β¦ cmpt 5224 βΆwf 6532 βcfv 6536 β©crio 7359 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 .rcmulr 17204 Scalarcsca 17206 Β·π cvsca 17207 0gc0g 17391 LSSumclsm 19551 LSpanclspn 20815 LVecclvec 20947 LSHypclsh 38357 LFnlclfn 38439 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cntz 19230 df-lsm 19553 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-drng 20586 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lvec 20948 df-lshyp 38359 df-lfl 38440 |
This theorem is referenced by: lshpkr 38499 lshpkrex 38500 dochflcl 40858 |
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