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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 34454 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
6 | eqid 2610 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
7 | dvh0g.o | . . . . 5 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
8 | 2, 3, 4, 6, 7 | tendo0cl 35096 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
9 | dvh0g.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2610 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
11 | eqid 2610 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
12 | eqid 2610 | . . . . 5 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈)) | |
13 | 3, 4, 6, 9, 10, 11, 12 | dvhopvadd 35400 | . . . 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 1327 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉) |
15 | f1oi 6086 | . . . . . 6 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
16 | f1of 6050 | . . . . . 6 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
17 | fcoi2 5992 | . . . . . 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 2610 | . . . . . . 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 35391 | . . . . . 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 6566 | . . . . 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 35097 | . . . . . 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 699 | . . . . 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 2644 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂) |
26 | 19, 25 | opeq12d 4348 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈(( I ↾ 𝐵) ∘ ( I ↾ 𝐵)), (𝑂(+g‘(Scalar‘𝑈))𝑂)〉 = 〈( I ↾ 𝐵), 𝑂〉) |
27 | 14, 26 | eqtrd 2644 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉) |
28 | 3, 9, 1 | dvhlmod 35417 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LMod) |
29 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
30 | 3, 4, 6, 9, 29 | dvhelvbasei 35395 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
31 | 1, 5, 8, 30 | syl12anc 1316 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 〈( I ↾ 𝐵), 𝑂〉 ∈ (Base‘𝑈)) |
32 | dvh0g.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
33 | 29, 11, 32 | lmod0vid 18718 | . . 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 691 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((〈( I ↾ 𝐵), 𝑂〉(+g‘𝑈)〈( I ↾ 𝐵), 𝑂〉) = 〈( I ↾ 𝐵), 𝑂〉 ↔ 0 = 〈( I ↾ 𝐵), 𝑂〉)) |
35 | 27, 34 | mpbid 221 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = 〈( I ↾ 𝐵), 𝑂〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 0gc0g 15923 LModclmod 18686 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 DVecHcdvh 35385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lvec 18924 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring 35063 df-dvech 35386 |
This theorem is referenced by: dvheveccl 35419 dib0 35471 dihmeetlem4preN 35613 dihmeetlem13N 35626 dihatlat 35641 dihpN 35643 |
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