![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapln1 | Structured version Visualization version GIF version |
Description: Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.) |
Ref | Expression |
---|---|
hdmapln1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapln1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapln1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapln1.p | ⊢ + = (+g‘𝑈) |
hdmapln1.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmapln1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapln1.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmapln1.q | ⊢ ⨣ = (+g‘𝑅) |
hdmapln1.m | ⊢ × = (.r‘𝑅) |
hdmapln1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapln1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapln1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmapln1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
hdmapln1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
hdmapln1.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
hdmapln1 | ⊢ (𝜑 → ((𝑆‘𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆‘𝑍)‘𝑋)) ⨣ ((𝑆‘𝑍)‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapln1.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapln1.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapln1.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 37639 | . 2 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | eqid 2772 | . . 3 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
6 | eqid 2772 | . . 3 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
7 | eqid 2772 | . . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
8 | hdmapln1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
9 | hdmapln1.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hdmapln1.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
11 | 1, 2, 8, 5, 6, 9, 3, 10 | hdmapcl 38359 | . . 3 ⊢ (𝜑 → (𝑆‘𝑍) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
12 | 1, 5, 6, 2, 7, 3, 11 | lcdvbaselfl 38124 | . 2 ⊢ (𝜑 → (𝑆‘𝑍) ∈ (LFnl‘𝑈)) |
13 | hdmapln1.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
14 | hdmapln1.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
15 | hdmapln1.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
16 | hdmapln1.p | . . 3 ⊢ + = (+g‘𝑈) | |
17 | hdmapln1.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
18 | hdmapln1.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
19 | hdmapln1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
20 | hdmapln1.q | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
21 | hdmapln1.m | . . 3 ⊢ × = (.r‘𝑅) | |
22 | 8, 16, 17, 18, 19, 20, 21, 7 | lfli 35590 | . 2 ⊢ ((𝑈 ∈ LMod ∧ (𝑆‘𝑍) ∈ (LFnl‘𝑈) ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝑆‘𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆‘𝑍)‘𝑋)) ⨣ ((𝑆‘𝑍)‘𝑌))) |
23 | 4, 12, 13, 14, 15, 22 | syl113anc 1362 | 1 ⊢ (𝜑 → ((𝑆‘𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆‘𝑍)‘𝑋)) ⨣ ((𝑆‘𝑍)‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 +gcplusg 16411 .rcmulr 16412 Scalarcsca 16414 ·𝑠 cvsca 16415 LModclmod 19346 LFnlclfn 35586 HLchlt 35879 LHypclh 36513 DVecHcdvh 37607 LCDualclcd 38115 HDMapchdma 38321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-riotaBAD 35482 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-undef 7735 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-0g 16561 df-mre 16705 df-mrc 16706 df-acs 16708 df-proset 17386 df-poset 17404 df-plt 17416 df-lub 17432 df-glb 17433 df-join 17434 df-meet 17435 df-p0 17497 df-p1 17498 df-lat 17504 df-clat 17566 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-grp 17884 df-minusg 17885 df-sbg 17886 df-subg 18050 df-cntz 18208 df-oppg 18235 df-lsm 18512 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-dvr 19146 df-drng 19217 df-lmod 19348 df-lss 19416 df-lsp 19456 df-lvec 19587 df-lsatoms 35505 df-lshyp 35506 df-lcv 35548 df-lfl 35587 df-lkr 35615 df-ldual 35653 df-oposet 35705 df-ol 35707 df-oml 35708 df-covers 35795 df-ats 35796 df-atl 35827 df-cvlat 35851 df-hlat 35880 df-llines 36027 df-lplanes 36028 df-lvols 36029 df-lines 36030 df-psubsp 36032 df-pmap 36033 df-padd 36325 df-lhyp 36517 df-laut 36518 df-ldil 36633 df-ltrn 36634 df-trl 36688 df-tgrp 37272 df-tendo 37284 df-edring 37286 df-dveca 37532 df-disoa 37558 df-dvech 37608 df-dib 37668 df-dic 37702 df-dih 37758 df-doch 37877 df-djh 37924 df-lcdual 38116 df-mapd 38154 df-hvmap 38286 df-hdmap1 38322 df-hdmap 38323 |
This theorem is referenced by: hdmapglem7b 38457 hlhilphllem 38488 |
Copyright terms: Public domain | W3C validator |