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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapip1 | Structured version Visualization version GIF version | ||
| Description: Construct a proportional vector π whose inner product with the original π equals one. (Contributed by NM, 13-Jun-2015.) |
| Ref | Expression |
|---|---|
| hdmapip1.h | β’ π» = (LHypβπΎ) |
| hdmapip1.u | β’ π = ((DVecHβπΎ)βπ) |
| hdmapip1.v | β’ π = (Baseβπ) |
| hdmapip1.t | β’ Β· = ( Β·π βπ) |
| hdmapip1.o | β’ 0 = (0gβπ) |
| hdmapip1.r | β’ π = (Scalarβπ) |
| hdmapip1.i | β’ 1 = (1rβπ ) |
| hdmapip1.n | β’ π = (invrβπ ) |
| hdmapip1.s | β’ π = ((HDMapβπΎ)βπ) |
| hdmapip1.k | β’ (π β (πΎ β HL β§ π β π»)) |
| hdmapip1.x | β’ (π β π β (π β { 0 })) |
| hdmapip1.y | β’ π = ((πβ((πβπ)βπ)) Β· π) |
| Ref | Expression |
|---|---|
| hdmapip1 | β’ (π β ((πβπ)βπ) = 1 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hdmapip1.y | . . 3 β’ π = ((πβ((πβπ)βπ)) Β· π) | |
| 2 | 1 | fveq2i 6905 | . 2 β’ ((πβπ)βπ) = ((πβπ)β((πβ((πβπ)βπ)) Β· π)) |
| 3 | hdmapip1.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 4 | hdmapip1.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
| 5 | hdmapip1.v | . . . 4 β’ π = (Baseβπ) | |
| 6 | hdmapip1.t | . . . 4 β’ Β· = ( Β·π βπ) | |
| 7 | hdmapip1.r | . . . 4 β’ π = (Scalarβπ) | |
| 8 | eqid 2728 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
| 9 | eqid 2728 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
| 10 | hdmapip1.s | . . . 4 β’ π = ((HDMapβπΎ)βπ) | |
| 11 | hdmapip1.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
| 12 | hdmapip1.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
| 13 | 12 | eldifad 3961 | . . . 4 β’ (π β π β π) |
| 14 | 3, 4, 11 | dvhlvec 40622 | . . . . . 6 β’ (π β π β LVec) |
| 15 | 7 | lvecdrng 21004 | . . . . . 6 β’ (π β LVec β π β DivRing) |
| 16 | 14, 15 | syl 17 | . . . . 5 β’ (π β π β DivRing) |
| 17 | 3, 4, 5, 7, 8, 10, 11, 13, 13 | hdmapipcl 41418 | . . . . 5 β’ (π β ((πβπ)βπ) β (Baseβπ )) |
| 18 | eldifsni 4798 | . . . . . . 7 β’ (π β (π β { 0 }) β π β 0 ) | |
| 19 | 12, 18 | syl 17 | . . . . . 6 β’ (π β π β 0 ) |
| 20 | hdmapip1.o | . . . . . . . 8 β’ 0 = (0gβπ) | |
| 21 | eqid 2728 | . . . . . . . 8 β’ (0gβπ ) = (0gβπ ) | |
| 22 | 3, 4, 5, 20, 7, 21, 10, 11, 13 | hdmapip0 41428 | . . . . . . 7 β’ (π β (((πβπ)βπ) = (0gβπ ) β π = 0 )) |
| 23 | 22 | necon3bid 2982 | . . . . . 6 β’ (π β (((πβπ)βπ) β (0gβπ ) β π β 0 )) |
| 24 | 19, 23 | mpbird 256 | . . . . 5 β’ (π β ((πβπ)βπ) β (0gβπ )) |
| 25 | hdmapip1.n | . . . . . 6 β’ π = (invrβπ ) | |
| 26 | 8, 21, 25 | drnginvrcl 20660 | . . . . 5 β’ ((π β DivRing β§ ((πβπ)βπ) β (Baseβπ ) β§ ((πβπ)βπ) β (0gβπ )) β (πβ((πβπ)βπ)) β (Baseβπ )) |
| 27 | 16, 17, 24, 26 | syl3anc 1368 | . . . 4 β’ (π β (πβ((πβπ)βπ)) β (Baseβπ )) |
| 28 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 13, 27 | hdmaplnm1 41422 | . . 3 β’ (π β ((πβπ)β((πβ((πβπ)βπ)) Β· π)) = ((πβ((πβπ)βπ))(.rβπ )((πβπ)βπ))) |
| 29 | hdmapip1.i | . . . . 5 β’ 1 = (1rβπ ) | |
| 30 | 8, 21, 9, 29, 25 | drnginvrl 20663 | . . . 4 β’ ((π β DivRing β§ ((πβπ)βπ) β (Baseβπ ) β§ ((πβπ)βπ) β (0gβπ )) β ((πβ((πβπ)βπ))(.rβπ )((πβπ)βπ)) = 1 ) |
| 31 | 16, 17, 24, 30 | syl3anc 1368 | . . 3 β’ (π β ((πβ((πβπ)βπ))(.rβπ )((πβπ)βπ)) = 1 ) |
| 32 | 28, 31 | eqtrd 2768 | . 2 β’ (π β ((πβπ)β((πβ((πβπ)βπ)) Β· π)) = 1 ) |
| 33 | 2, 32 | eqtrid 2780 | 1 β’ (π β ((πβπ)βπ) = 1 ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 {csn 4632 βcfv 6553 (class class class)co 7426 Basecbs 17189 .rcmulr 17243 Scalarcsca 17245 Β·π cvsca 17246 0gc0g 17430 1rcur 20135 invrcinvr 20340 DivRingcdr 20638 LVecclvec 21001 HLchlt 38862 LHypclh 39497 DVecHcdvh 40591 HDMapchdma 41305 |
| 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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-riotaBAD 38465 |
| 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-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 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-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-undef 8287 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-0g 17432 df-mre 17575 df-mrc 17576 df-acs 17578 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cntz 19282 df-oppg 19311 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-drng 20640 df-lmod 20759 df-lss 20830 df-lsp 20870 df-lvec 21002 df-lsatoms 38488 df-lshyp 38489 df-lcv 38531 df-lfl 38570 df-lkr 38598 df-ldual 38636 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-llines 39011 df-lplanes 39012 df-lvols 39013 df-lines 39014 df-psubsp 39016 df-pmap 39017 df-padd 39309 df-lhyp 39501 df-laut 39502 df-ldil 39617 df-ltrn 39618 df-trl 39672 df-tgrp 40256 df-tendo 40268 df-edring 40270 df-dveca 40516 df-disoa 40542 df-dvech 40592 df-dib 40652 df-dic 40686 df-dih 40742 df-doch 40861 df-djh 40908 df-lcdual 41100 df-mapd 41138 df-hvmap 41270 df-hdmap1 41306 df-hdmap 41307 |
| This theorem is referenced by: hgmapvvlem3 41438 |
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