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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 2734 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | tendo0.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendo0.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | eqid 2734 | . 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 40132 | . . . 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 40768 | . . 3 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
12 | 9, 11 | fmptd 7133 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂:𝑇⟶𝑇) |
13 | 7, 2, 3, 5, 10 | tendo0co2 40770 | . 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 40771 | . 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 40744 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ↦ cmpt 5230 I cid 5581 ↾ cres 5690 ‘cfv 6562 Basecbs 17244 lecple 17304 HLchlt 39331 LHypclh 39966 LTrncltrn 40083 trLctrl 40140 TEndoctendo 40734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-undef 8296 df-map 8866 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tendo 40737 |
This theorem is referenced by: tendo0pl 40773 tendo0plr 40774 tendoipl 40779 tendoid0 40807 tendo0mul 40808 tendo0mulr 40809 tendoex 40957 cdleml5N 40962 erngdvlem1 40970 erngdvlem4 40973 erng0g 40976 erngdvlem1-rN 40978 erngdvlem4-rN 40981 dvh0g 41093 dvhopN 41098 dib1dim 41147 dib1dim2 41150 dibss 41151 diblss 41152 diblsmopel 41153 dicn0 41174 cdlemn4 41180 cdlemn4a 41181 cdlemn6 41184 dihopelvalcpre 41230 dihmeetlem4preN 41288 dihatlat 41316 dihatexv 41320 |
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