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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvh0g | Structured version Visualization version GIF version |
Description: The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
dvh0g.b | ⊢ 𝐵 = (Base‘𝐾) |
dvh0g.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvh0g.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvh0g.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvh0g.z | ⊢ 0 = (0g‘𝑈) |
dvh0g.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
dvh0g | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = 〈( I ↾ 𝐵), 𝑂〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dvh0g.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | dvh0g.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dvh0g.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | idltrn 36299 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
6 | eqid 2777 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
7 | dvh0g.o | . . . . 5 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
8 | 2, 3, 4, 6, 7 | tendo0cl 36939 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
9 | dvh0g.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2777 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
11 | eqid 2777 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
12 | eqid 2777 | . . . . 5 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈)) | |
13 | 3, 4, 6, 9, 10, 11, 12 | dvhopvadd 37242 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉) |
14 | 1, 5, 8, 5, 8, 13 | syl122anc 1447 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉) |
15 | f1oi 6428 | . . . . . 6 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
16 | f1of 6391 | . . . . . 6 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
17 | fcoi2 6329 | . . . . . 6 ⊢ (( I ↾ 𝐵):𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ ( I ↾ 𝐵)) = ( I ↾ 𝐵)) | |
18 | 15, 16, 17 | mp2b 10 | . . . . 5 ⊢ (( I ↾ 𝐵) ∘ ( I ↾ 𝐵)) = ( I ↾ 𝐵) |
19 | 18 | a1i 11 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝐵) ∘ ( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
20 | eqid 2777 | . . . . . . 7 ⊢ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
21 | 3, 4, 6, 9, 10, 20, 12 | dvhfplusr 37233 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))) |
22 | 21 | oveqd 6939 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))𝑂)) |
23 | 2, 3, 4, 6, 7, 20 | tendo0pl 36940 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))𝑂) = 𝑂) |
24 | 8, 23 | mpdan 677 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))𝑂) = 𝑂) |
25 | 22, 24 | eqtrd 2813 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂) |
26 | 19, 25 | opeq12d 4644 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉 = 〈( I ↾ 𝐵), 𝑂〉) |
27 | 14, 26 | eqtrd 2813 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉) |
28 | 3, 9, 1 | dvhlmod 37259 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LMod) |
29 | eqid 2777 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
30 | 3, 4, 6, 9, 29 | dvhelvbasei 37237 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
31 | 1, 5, 8, 30 | syl12anc 827 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
32 | dvh0g.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
33 | 29, 11, 32 | lmod0vid 19287 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) → ((〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉 ↔ 0 = 〈( I ↾ 𝐵), 𝑂〉)) |
34 | 28, 31, 33 | syl2anc 579 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉 ↔ 0 = 〈( I ↾ 𝐵), 𝑂〉)) |
35 | 27, 34 | mpbid 224 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = 〈( I ↾ 𝐵), 𝑂〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 〈cop 4403 ↦ cmpt 4965 I cid 5260 ↾ cres 5357 ∘ ccom 5359 ⟶wf 6131 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 Basecbs 16255 +gcplusg 16338 Scalarcsca 16341 0gc0g 16486 LModclmod 19255 HLchlt 35499 LHypclh 36133 LTrncltrn 36250 TEndoctendo 36901 DVecHcdvh 37227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35102 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lvec 19498 df-oposet 35325 df-ol 35327 df-oml 35328 df-covers 35415 df-ats 35416 df-atl 35447 df-cvlat 35471 df-hlat 35500 df-llines 35647 df-lplanes 35648 df-lvols 35649 df-lines 35650 df-psubsp 35652 df-pmap 35653 df-padd 35945 df-lhyp 36137 df-laut 36138 df-ldil 36253 df-ltrn 36254 df-trl 36308 df-tendo 36904 df-edring 36906 df-dvech 37228 |
This theorem is referenced by: dvheveccl 37261 dib0 37313 dihmeetlem4preN 37455 dihmeetlem13N 37468 dihatlat 37483 dihpN 37485 |
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