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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 39010 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
6 | eqid 2733 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
7 | dvh0g.o | . . . . 5 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
8 | 2, 3, 4, 6, 7 | tendo0cl 39650 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
9 | dvh0g.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2733 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
11 | eqid 2733 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
12 | eqid 2733 | . . . . 5 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈)) | |
13 | 3, 4, 6, 9, 10, 11, 12 | dvhopvadd 39953 | . . . 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 1380 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉) |
15 | f1oi 6869 | . . . . . 6 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
16 | f1of 6831 | . . . . . 6 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
17 | fcoi2 6764 | . . . . . 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 2733 | . . . . . . 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 39944 | . . . . . 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 7423 | . . . . 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 39651 | . . . . . 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 686 | . . . . 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 2773 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂) |
26 | 19, 25 | opeq12d 4881 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉 = 〈( I ↾ 𝐵), 𝑂〉) |
27 | 14, 26 | eqtrd 2773 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉) |
28 | 3, 9, 1 | dvhlmod 39970 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LMod) |
29 | eqid 2733 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
30 | 3, 4, 6, 9, 29 | dvhelvbasei 39948 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
31 | 1, 5, 8, 30 | syl12anc 836 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
32 | dvh0g.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
33 | 29, 11, 32 | lmod0vid 20497 | . . 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 585 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉 ↔ 0 = 〈( I ↾ 𝐵), 𝑂〉)) |
35 | 27, 34 | mpbid 231 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = 〈( I ↾ 𝐵), 𝑂〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 〈cop 4634 ↦ cmpt 5231 I cid 5573 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 Basecbs 17141 +gcplusg 17194 Scalarcsca 17197 0gc0g 17382 LModclmod 20464 HLchlt 38209 LHypclh 38844 LTrncltrn 38961 TEndoctendo 39612 DVecHcdvh 39938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 37812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-0g 17384 df-proset 18245 df-poset 18263 df-plt 18280 df-lub 18296 df-glb 18297 df-join 18298 df-meet 18299 df-p0 18375 df-p1 18376 df-lat 18382 df-clat 18449 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-minusg 18820 df-mgp 19983 df-ur 20000 df-ring 20052 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-dvr 20208 df-drng 20310 df-lmod 20466 df-lvec 20707 df-oposet 38035 df-ol 38037 df-oml 38038 df-covers 38125 df-ats 38126 df-atl 38157 df-cvlat 38181 df-hlat 38210 df-llines 38358 df-lplanes 38359 df-lvols 38360 df-lines 38361 df-psubsp 38363 df-pmap 38364 df-padd 38656 df-lhyp 38848 df-laut 38849 df-ldil 38964 df-ltrn 38965 df-trl 39019 df-tendo 39615 df-edring 39617 df-dvech 39939 |
This theorem is referenced by: dvheveccl 39972 dib0 40024 dihmeetlem4preN 40166 dihmeetlem13N 40179 dihatlat 40194 dihpN 40196 |
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