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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0cl | Structured version Visualization version GIF version |
Description: The additive identity is a trace-preserving endormorphism. (Contributed by NM, 12-Jun-2013.) |
Ref | Expression |
---|---|
tendo0.b | ⊢ 𝐵 = (Base‘𝐾) |
tendo0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendo0.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendo0.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
tendo0.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
tendo0cl | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | tendo0.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendo0.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | eqid 2731 | . 2 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
5 | tendo0.e | . 2 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
6 | id 22 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | tendo0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 7, 2, 3 | idltrn 39485 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
9 | 8 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → ( I ↾ 𝐵) ∈ 𝑇) |
10 | tendo0.o | . . . 4 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
11 | 10 | tendo0cbv 40121 | . . 3 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
12 | 9, 11 | fmptd 7115 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂:𝑇⟶𝑇) |
13 | 7, 2, 3, 5, 10 | tendo0co2 40123 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ ℎ ∈ 𝑇) → (𝑂‘(𝑔 ∘ ℎ)) = ((𝑂‘𝑔) ∘ (𝑂‘ℎ))) |
14 | 7, 2, 3, 5, 10, 1, 4 | tendo0tp 40124 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘(𝑂‘𝑔))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔)) |
15 | 1, 2, 3, 4, 5, 6, 12, 13, 14 | istendod 40097 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ↦ cmpt 5231 I cid 5573 ↾ cres 5678 ‘cfv 6543 Basecbs 17151 lecple 17211 HLchlt 38684 LHypclh 39319 LTrncltrn 39436 trLctrl 39493 TEndoctendo 40087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-riotaBAD 38287 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-undef 8264 df-map 8828 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-oposet 38510 df-ol 38512 df-oml 38513 df-covers 38600 df-ats 38601 df-atl 38632 df-cvlat 38656 df-hlat 38685 df-llines 38833 df-lplanes 38834 df-lvols 38835 df-lines 38836 df-psubsp 38838 df-pmap 38839 df-padd 39131 df-lhyp 39323 df-laut 39324 df-ldil 39439 df-ltrn 39440 df-trl 39494 df-tendo 40090 |
This theorem is referenced by: tendo0pl 40126 tendo0plr 40127 tendoipl 40132 tendoid0 40160 tendo0mul 40161 tendo0mulr 40162 tendoex 40310 cdleml5N 40315 erngdvlem1 40323 erngdvlem4 40326 erng0g 40329 erngdvlem1-rN 40331 erngdvlem4-rN 40334 dvh0g 40446 dvhopN 40451 dib1dim 40500 dib1dim2 40503 dibss 40504 diblss 40505 diblsmopel 40506 dicn0 40527 cdlemn4 40533 cdlemn4a 40534 cdlemn6 40537 dihopelvalcpre 40583 dihmeetlem4preN 40641 dihatlat 40669 dihatexv 40673 |
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