![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapinvlem1 | Structured version Visualization version GIF version |
Description: Line 27 in [Baer] p. 110. We use 𝐶 for Baer's u. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 41441. Our ((𝑆‘𝐸)‘𝐶) means the inner product 〈𝐶, 𝐸〉 i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.) |
Ref | Expression |
---|---|
hdmapinvlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapinvlem1.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapinvlem1.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
hdmapinvlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapinvlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapinvlem1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapinvlem1.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmapinvlem1.t | ⊢ · = (.r‘𝑅) |
hdmapinvlem1.z | ⊢ 0 = (0g‘𝑅) |
hdmapinvlem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapinvlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapinvlem1.c | ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) |
Ref | Expression |
---|---|
hdmapinvlem1 | ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapinvlem1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) | |
2 | hdmapinvlem1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | hdmapinvlem1.o | . . . 4 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
4 | hdmapinvlem1.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | hdmapinvlem1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
6 | eqid 2725 | . . . 4 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2725 | . . . 4 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | hdmapinvlem1.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
9 | hdmapinvlem1.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | eqid 2725 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
12 | eqid 2725 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
13 | hdmapinvlem1.e | . . . . . 6 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
14 | 2, 10, 11, 4, 5, 12, 13, 9 | dvheveccl 40717 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
15 | 14 | eldifad 3956 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
16 | 2, 3, 4, 5, 6, 7, 8, 9, 15 | hdmaplkr 41518 | . . 3 ⊢ (𝜑 → ((LKer‘𝑈)‘(𝑆‘𝐸)) = (𝑂‘{𝐸})) |
17 | 1, 16 | eleqtrrd 2828 | . 2 ⊢ (𝜑 → 𝐶 ∈ ((LKer‘𝑈)‘(𝑆‘𝐸))) |
18 | hdmapinvlem1.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
19 | hdmapinvlem1.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
20 | 2, 4, 9 | dvhlmod 40715 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
21 | eqid 2725 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
22 | eqid 2725 | . . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
23 | 2, 4, 5, 21, 22, 8, 9, 15 | hdmapcl 41435 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
24 | 2, 21, 22, 4, 6, 9, 23 | lcdvbaselfl 41200 | . . 3 ⊢ (𝜑 → (𝑆‘𝐸) ∈ (LFnl‘𝑈)) |
25 | 15 | snssd 4814 | . . . . 5 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
26 | 2, 4, 5, 3 | dochssv 40960 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉) |
27 | 9, 25, 26 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉) |
28 | 27, 1 | sseldd 3977 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
29 | 5, 18, 19, 6, 7, 20, 24, 28 | ellkr2 38695 | . 2 ⊢ (𝜑 → (𝐶 ∈ ((LKer‘𝑈)‘(𝑆‘𝐸)) ↔ ((𝑆‘𝐸)‘𝐶) = 0 )) |
30 | 17, 29 | mpbid 231 | 1 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 {csn 4630 〈cop 4636 I cid 5575 ↾ cres 5680 ‘cfv 6549 Basecbs 17188 .rcmulr 17242 Scalarcsca 17244 0gc0g 17429 LModclmod 20760 LFnlclfn 38661 LKerclk 38689 HLchlt 38954 LHypclh 39589 LTrncltrn 39706 DVecHcdvh 40683 ocHcoch 40952 LCDualclcd 41191 HDMapchdma 41397 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38557 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-sca 17257 df-vsca 17258 df-0g 17431 df-mre 17574 df-mrc 17575 df-acs 17577 df-proset 18295 df-poset 18313 df-plt 18330 df-lub 18346 df-glb 18347 df-join 18348 df-meet 18349 df-p0 18425 df-p1 18426 df-lat 18432 df-clat 18499 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19091 df-cntz 19285 df-oppg 19314 df-lsm 19608 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-dvr 20357 df-drng 20643 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-lsatoms 38580 df-lshyp 38581 df-lcv 38623 df-lfl 38662 df-lkr 38690 df-ldual 38728 df-oposet 38780 df-ol 38782 df-oml 38783 df-covers 38870 df-ats 38871 df-atl 38902 df-cvlat 38926 df-hlat 38955 df-llines 39103 df-lplanes 39104 df-lvols 39105 df-lines 39106 df-psubsp 39108 df-pmap 39109 df-padd 39401 df-lhyp 39593 df-laut 39594 df-ldil 39709 df-ltrn 39710 df-trl 39764 df-tgrp 40348 df-tendo 40360 df-edring 40362 df-dveca 40608 df-disoa 40634 df-dvech 40684 df-dib 40744 df-dic 40778 df-dih 40834 df-doch 40953 df-djh 41000 df-lcdual 41192 df-mapd 41230 df-hvmap 41362 df-hdmap1 41398 df-hdmap 41399 |
This theorem is referenced by: hdmapinvlem2 41524 hdmapinvlem3 41525 hdmapinvlem4 41526 hdmapglem7b 41533 |
Copyright terms: Public domain | W3C validator |