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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaplna2 | Structured version Visualization version GIF version |
Description: Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.) |
Ref | Expression |
---|---|
hdmaplna2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmaplna2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmaplna2.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmaplna2.p | ⊢ + = (+g‘𝑈) |
hdmaplna2.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmaplna2.q | ⊢ ⨣ = (+g‘𝑅) |
hdmaplna2.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmaplna2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmaplna2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmaplna2.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
hdmaplna2.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmaplna2 | ⊢ (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆‘𝑌)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaplna2.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmaplna2.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmaplna2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmaplna2.p | . . . 4 ⊢ + = (+g‘𝑈) | |
5 | eqid 2739 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
6 | eqid 2739 | . . . 4 ⊢ (+g‘((LCDual‘𝐾)‘𝑊)) = (+g‘((LCDual‘𝐾)‘𝑊)) | |
7 | hdmaplna2.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
8 | hdmaplna2.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | hdmaplna2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
10 | hdmaplna2.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | hdmapadd 39503 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑌 + 𝑍)) = ((𝑆‘𝑌)(+g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑍))) |
12 | 11 | fveq1d 6679 | . 2 ⊢ (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆‘𝑌)(+g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑍))‘𝑋)) |
13 | hdmaplna2.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
14 | hdmaplna2.q | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
15 | eqid 2739 | . . 3 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
16 | 1, 2, 3, 5, 15, 7, 8, 9 | hdmapcl 39490 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
17 | 1, 2, 3, 5, 15, 7, 8, 10 | hdmapcl 39490 | . . 3 ⊢ (𝜑 → (𝑆‘𝑍) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
18 | hdmaplna2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
19 | 1, 2, 3, 13, 14, 5, 15, 6, 8, 16, 17, 18 | lcdvaddval 39258 | . 2 ⊢ (𝜑 → (((𝑆‘𝑌)(+g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑍))‘𝑋) = (((𝑆‘𝑌)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑋))) |
20 | 12, 19 | eqtrd 2774 | 1 ⊢ (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆‘𝑌)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 +gcplusg 16671 Scalarcsca 16674 HLchlt 37010 LHypclh 37644 DVecHcdvh 38738 LCDualclcd 39246 HDMapchdma 39452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-riotaBAD 36613 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-ot 4526 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-tpos 7924 df-undef 7971 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-0g 16821 df-mre 16963 df-mrc 16964 df-acs 16966 df-proset 17657 df-poset 17675 df-plt 17687 df-lub 17703 df-glb 17704 df-join 17705 df-meet 17706 df-p0 17768 df-p1 17769 df-lat 17775 df-clat 17837 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-grp 18225 df-minusg 18226 df-sbg 18227 df-subg 18397 df-cntz 18568 df-oppg 18595 df-lsm 18882 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-ring 19421 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-invr 19547 df-dvr 19558 df-drng 19626 df-lmod 19758 df-lss 19826 df-lsp 19866 df-lvec 19997 df-lsatoms 36636 df-lshyp 36637 df-lcv 36679 df-lfl 36718 df-lkr 36746 df-ldual 36784 df-oposet 36836 df-ol 36838 df-oml 36839 df-covers 36926 df-ats 36927 df-atl 36958 df-cvlat 36982 df-hlat 37011 df-llines 37158 df-lplanes 37159 df-lvols 37160 df-lines 37161 df-psubsp 37163 df-pmap 37164 df-padd 37456 df-lhyp 37648 df-laut 37649 df-ldil 37764 df-ltrn 37765 df-trl 37819 df-tgrp 38403 df-tendo 38415 df-edring 38417 df-dveca 38663 df-disoa 38689 df-dvech 38739 df-dib 38799 df-dic 38833 df-dih 38889 df-doch 39008 df-djh 39055 df-lcdual 39247 df-mapd 39285 df-hvmap 39417 df-hdmap1 39453 df-hdmap 39454 |
This theorem is referenced by: hdmapgln2 39572 hdmapinvlem4 39581 |
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