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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 2726 | . . . . . . . 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 480 | . . . . . . . 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 614 | . . . . . . . . 9 β’ ((π β§ π β 0 ) β (π β π β§ π β 0 )) |
10 | eldifsn 4785 | . . . . . . . . 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 40777 | . . . . . . 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 2726 | . . . . . . . . . . . 12 β’ (LFnlβπ) = (LFnlβπ) | |
16 | eqid 2726 | . . . . . . . . . . . 12 β’ (LKerβπ) = (LKerβπ) | |
17 | 1, 3, 6 | dvhlmod 40494 | . . . . . . . . . . . 12 β’ (π β π β LMod) |
18 | eqid 2726 | . . . . . . . . . . . . 13 β’ ((LCDualβπΎ)βπ) = ((LCDualβπΎ)βπ) | |
19 | eqid 2726 | . . . . . . . . . . . . 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 41214 | . . . . . . . . . . . . 13 β’ (π β (πβπ) β (Baseβ((LCDualβπΎ)βπ))) |
22 | 1, 18, 19, 3, 15, 6, 21 | lcdvbaselfl 40979 | . . . . . . . . . . . 12 β’ (π β (πβπ) β (LFnlβπ)) |
23 | 4, 13, 14, 15, 16, 17, 22, 8 | ellkr2 38474 | . . . . . . . . . . 11 β’ (π β (π β ((LKerβπ)β(πβπ)) β ((πβπ)βπ) = π)) |
24 | 23 | biimpar 477 | . . . . . . . . . 10 β’ ((π β§ ((πβπ)βπ) = π) β π β ((LKerβπ)β(πβπ))) |
25 | 1, 2, 3, 4, 15, 16, 20, 6, 8 | hdmaplkr 41297 | . . . . . . . . . . 11 β’ (π β ((LKerβπ)β(πβπ)) = (((ocHβπΎ)βπ)β{π})) |
26 | 25 | adantr 480 | . . . . . . . . . 10 β’ ((π β§ ((πβπ)βπ) = π) β ((LKerβπ)β(πβπ)) = (((ocHβπΎ)βπ)β{π})) |
27 | 24, 26 | eleqtrd 2829 | . . . . . . . . 9 β’ ((π β§ ((πβπ)βπ) = π) β π β (((ocHβπΎ)βπ)β{π})) |
28 | 27 | ex 412 | . . . . . . . 8 β’ (π β (((πβπ)βπ) = π β π β (((ocHβπΎ)βπ)β{π}))) |
29 | 28 | adantr 480 | . . . . . . 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 2941 | . . . . 5 β’ ((π β§ π β 0 ) β ((πβπ)βπ) β π) |
32 | 31 | ex 412 | . . . 4 β’ (π β (π β 0 β ((πβπ)βπ) β π)) |
33 | 32 | necon4d 2958 | . . 3 β’ (π β (((πβπ)βπ) = π β π = 0 )) |
34 | 33 | imp 406 | . 2 β’ ((π β§ ((πβπ)βπ) = π) β π = 0 ) |
35 | fveq2 6885 | . . 3 β’ (π = 0 β ((πβπ)βπ) = ((πβπ)β 0 )) | |
36 | 13, 14, 5, 15 | lfl0 38448 | . . . 4 β’ ((π β LMod β§ (πβπ) β (LFnlβπ)) β ((πβπ)β 0 ) = π) |
37 | 17, 22, 36 | syl2anc 583 | . . 3 β’ (π β ((πβπ)β 0 ) = π) |
38 | 35, 37 | sylan9eqr 2788 | . 2 β’ ((π β§ π = 0 ) β ((πβπ)βπ) = π) |
39 | 34, 38 | impbida 798 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 β cdif 3940 {csn 4623 βcfv 6537 Basecbs 17153 Scalarcsca 17209 0gc0g 17394 LModclmod 20706 LFnlclfn 38440 LKerclk 38468 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 ocHcoch 40731 LCDualclcd 40970 HDMapchdma 41176 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336 |
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-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-oppg 19262 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lcv 38402 df-lfl 38441 df-lkr 38469 df-ldual 38507 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779 df-lcdual 40971 df-mapd 41009 df-hvmap 41141 df-hdmap1 41177 df-hdmap 41178 |
This theorem is referenced by: hdmapip1 41300 hgmapvvlem3 41309 hlhilphllem 41347 |
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