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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapip0 | Structured version Visualization version GIF version |
Description: Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.) |
Ref | Expression |
---|---|
hdmapip0.h | β’ π» = (LHypβπΎ) |
hdmapip0.u | β’ π = ((DVecHβπΎ)βπ) |
hdmapip0.v | β’ π = (Baseβπ) |
hdmapip0.o | β’ 0 = (0gβπ) |
hdmapip0.r | β’ π = (Scalarβπ) |
hdmapip0.z | β’ π = (0gβπ ) |
hdmapip0.s | β’ π = ((HDMapβπΎ)βπ) |
hdmapip0.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmapip0.x | β’ (π β π β π) |
Ref | Expression |
---|---|
hdmapip0 | β’ (π β (((πβπ)βπ) = π β π = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapip0.h | . . . . . . . 8 β’ π» = (LHypβπΎ) | |
2 | eqid 2737 | . . . . . . . 8 β’ ((ocHβπΎ)βπ) = ((ocHβπΎ)βπ) | |
3 | hdmapip0.u | . . . . . . . 8 β’ π = ((DVecHβπΎ)βπ) | |
4 | hdmapip0.v | . . . . . . . 8 β’ π = (Baseβπ) | |
5 | hdmapip0.o | . . . . . . . 8 β’ 0 = (0gβπ) | |
6 | hdmapip0.k | . . . . . . . . 9 β’ (π β (πΎ β HL β§ π β π»)) | |
7 | 6 | adantr 482 | . . . . . . . 8 β’ ((π β§ π β 0 ) β (πΎ β HL β§ π β π»)) |
8 | hdmapip0.x | . . . . . . . . . 10 β’ (π β π β π) | |
9 | 8 | anim1i 616 | . . . . . . . . 9 β’ ((π β§ π β 0 ) β (π β π β§ π β 0 )) |
10 | eldifsn 4748 | . . . . . . . . 9 β’ (π β (π β { 0 }) β (π β π β§ π β 0 )) | |
11 | 9, 10 | sylibr 233 | . . . . . . . 8 β’ ((π β§ π β 0 ) β π β (π β { 0 })) |
12 | 1, 2, 3, 4, 5, 7, 11 | dochnel 39859 | . . . . . . 7 β’ ((π β§ π β 0 ) β Β¬ π β (((ocHβπΎ)βπ)β{π})) |
13 | hdmapip0.r | . . . . . . . . . . . 12 β’ π = (Scalarβπ) | |
14 | hdmapip0.z | . . . . . . . . . . . 12 β’ π = (0gβπ ) | |
15 | eqid 2737 | . . . . . . . . . . . 12 β’ (LFnlβπ) = (LFnlβπ) | |
16 | eqid 2737 | . . . . . . . . . . . 12 β’ (LKerβπ) = (LKerβπ) | |
17 | 1, 3, 6 | dvhlmod 39576 | . . . . . . . . . . . 12 β’ (π β π β LMod) |
18 | eqid 2737 | . . . . . . . . . . . . 13 β’ ((LCDualβπΎ)βπ) = ((LCDualβπΎ)βπ) | |
19 | eqid 2737 | . . . . . . . . . . . . 13 β’ (Baseβ((LCDualβπΎ)βπ)) = (Baseβ((LCDualβπΎ)βπ)) | |
20 | hdmapip0.s | . . . . . . . . . . . . . 14 β’ π = ((HDMapβπΎ)βπ) | |
21 | 1, 3, 4, 18, 19, 20, 6, 8 | hdmapcl 40296 | . . . . . . . . . . . . 13 β’ (π β (πβπ) β (Baseβ((LCDualβπΎ)βπ))) |
22 | 1, 18, 19, 3, 15, 6, 21 | lcdvbaselfl 40061 | . . . . . . . . . . . 12 β’ (π β (πβπ) β (LFnlβπ)) |
23 | 4, 13, 14, 15, 16, 17, 22, 8 | ellkr2 37556 | . . . . . . . . . . 11 β’ (π β (π β ((LKerβπ)β(πβπ)) β ((πβπ)βπ) = π)) |
24 | 23 | biimpar 479 | . . . . . . . . . 10 β’ ((π β§ ((πβπ)βπ) = π) β π β ((LKerβπ)β(πβπ))) |
25 | 1, 2, 3, 4, 15, 16, 20, 6, 8 | hdmaplkr 40379 | . . . . . . . . . . 11 β’ (π β ((LKerβπ)β(πβπ)) = (((ocHβπΎ)βπ)β{π})) |
26 | 25 | adantr 482 | . . . . . . . . . 10 β’ ((π β§ ((πβπ)βπ) = π) β ((LKerβπ)β(πβπ)) = (((ocHβπΎ)βπ)β{π})) |
27 | 24, 26 | eleqtrd 2840 | . . . . . . . . 9 β’ ((π β§ ((πβπ)βπ) = π) β π β (((ocHβπΎ)βπ)β{π})) |
28 | 27 | ex 414 | . . . . . . . 8 β’ (π β (((πβπ)βπ) = π β π β (((ocHβπΎ)βπ)β{π}))) |
29 | 28 | adantr 482 | . . . . . . 7 β’ ((π β§ π β 0 ) β (((πβπ)βπ) = π β π β (((ocHβπΎ)βπ)β{π}))) |
30 | 12, 29 | mtod 197 | . . . . . 6 β’ ((π β§ π β 0 ) β Β¬ ((πβπ)βπ) = π) |
31 | 30 | neqned 2951 | . . . . 5 β’ ((π β§ π β 0 ) β ((πβπ)βπ) β π) |
32 | 31 | ex 414 | . . . 4 β’ (π β (π β 0 β ((πβπ)βπ) β π)) |
33 | 32 | necon4d 2968 | . . 3 β’ (π β (((πβπ)βπ) = π β π = 0 )) |
34 | 33 | imp 408 | . 2 β’ ((π β§ ((πβπ)βπ) = π) β π = 0 ) |
35 | fveq2 6843 | . . 3 β’ (π = 0 β ((πβπ)βπ) = ((πβπ)β 0 )) | |
36 | 13, 14, 5, 15 | lfl0 37530 | . . . 4 β’ ((π β LMod β§ (πβπ) β (LFnlβπ)) β ((πβπ)β 0 ) = π) |
37 | 17, 22, 36 | syl2anc 585 | . . 3 β’ (π β ((πβπ)β 0 ) = π) |
38 | 35, 37 | sylan9eqr 2799 | . 2 β’ ((π β§ π = 0 ) β ((πβπ)βπ) = π) |
39 | 34, 38 | impbida 800 | 1 β’ (π β (((πβπ)βπ) = π β π = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 β cdif 3908 {csn 4587 βcfv 6497 Basecbs 17084 Scalarcsca 17137 0gc0g 17322 LModclmod 20325 LFnlclfn 37522 LKerclk 37550 HLchlt 37815 LHypclh 38450 DVecHcdvh 39544 ocHcoch 39813 LCDualclcd 40052 HDMapchdma 40258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-riotaBAD 37418 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-0g 17324 df-mre 17467 df-mrc 17468 df-acs 17470 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-submnd 18603 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-cntz 19098 df-oppg 19125 df-lsm 19419 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-drng 20188 df-lmod 20327 df-lss 20396 df-lsp 20436 df-lvec 20567 df-lsatoms 37441 df-lshyp 37442 df-lcv 37484 df-lfl 37523 df-lkr 37551 df-ldual 37589 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-tgrp 39209 df-tendo 39221 df-edring 39223 df-dveca 39469 df-disoa 39495 df-dvech 39545 df-dib 39605 df-dic 39639 df-dih 39695 df-doch 39814 df-djh 39861 df-lcdual 40053 df-mapd 40091 df-hvmap 40223 df-hdmap1 40259 df-hdmap 40260 |
This theorem is referenced by: hdmapip1 40382 hgmapvvlem3 40391 hlhilphllem 40429 |
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