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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 35939 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
6 | eqid 2760 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
7 | dvh0g.o | . . . . 5 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
8 | 2, 3, 4, 6, 7 | tendo0cl 36580 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
9 | dvh0g.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2760 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
11 | eqid 2760 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
12 | eqid 2760 | . . . . 5 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈)) | |
13 | 3, 4, 6, 9, 10, 11, 12 | dvhopvadd 36884 | . . . 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 1486 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉) |
15 | f1oi 6335 | . . . . . 6 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
16 | f1of 6298 | . . . . . 6 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
17 | fcoi2 6240 | . . . . . 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 2760 | . . . . . . 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 36875 | . . . . . 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 6830 | . . . . 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 36581 | . . . . . 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 705 | . . . . 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 2794 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂) |
26 | 19, 25 | opeq12d 4561 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉 = 〈( I ↾ 𝐵), 𝑂〉) |
27 | 14, 26 | eqtrd 2794 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉) |
28 | 3, 9, 1 | dvhlmod 36901 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LMod) |
29 | eqid 2760 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
30 | 3, 4, 6, 9, 29 | dvhelvbasei 36879 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
31 | 1, 5, 8, 30 | syl12anc 1475 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
32 | dvh0g.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
33 | 29, 11, 32 | lmod0vid 19097 | . . 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 696 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉 ↔ 0 = 〈( I ↾ 𝐵), 𝑂〉)) |
35 | 27, 34 | mpbid 222 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = 〈( I ↾ 𝐵), 𝑂〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 ↦ cmpt 4881 I cid 5173 ↾ cres 5268 ∘ ccom 5270 ⟶wf 6045 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6813 ↦ cmpt2 6815 Basecbs 16059 +gcplusg 16143 Scalarcsca 16146 0gc0g 16302 LModclmod 19065 HLchlt 35140 LHypclh 35773 LTrncltrn 35890 TEndoctendo 36542 DVecHcdvh 36869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-riotaBAD 34742 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-tpos 7521 df-undef 7568 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-0g 16304 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 df-mgp 18690 df-ur 18702 df-ring 18749 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-drng 18951 df-lmod 19067 df-lvec 19305 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 df-lvols 35289 df-lines 35290 df-psubsp 35292 df-pmap 35293 df-padd 35585 df-lhyp 35777 df-laut 35778 df-ldil 35893 df-ltrn 35894 df-trl 35949 df-tendo 36545 df-edring 36547 df-dvech 36870 |
This theorem is referenced by: dvheveccl 36903 dib0 36955 dihmeetlem4preN 37097 dihmeetlem13N 37110 dihatlat 37125 dihpN 37127 |
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