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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-perserving 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 2799 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | tendo0.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendo0.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | eqid 2799 | . 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 36171 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
9 | 8 | adantr 473 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → ( I ↾ 𝐵) ∈ 𝑇) |
10 | tendo0.o | . . . 4 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
11 | 10 | tendo0cbv 36807 | . . 3 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
12 | 9, 11 | fmptd 6610 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂:𝑇⟶𝑇) |
13 | 7, 2, 3, 5, 10 | tendo0co2 36809 | . 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 36810 | . 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 36783 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ↦ cmpt 4922 I cid 5219 ↾ cres 5314 ‘cfv 6101 Basecbs 16184 lecple 16274 HLchlt 35371 LHypclh 36005 LTrncltrn 36122 trLctrl 36179 TEndoctendo 36773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-riotaBAD 34974 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-undef 7637 df-map 8097 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-llines 35519 df-lplanes 35520 df-lvols 35521 df-lines 35522 df-psubsp 35524 df-pmap 35525 df-padd 35817 df-lhyp 36009 df-laut 36010 df-ldil 36125 df-ltrn 36126 df-trl 36180 df-tendo 36776 |
This theorem is referenced by: tendo0pl 36812 tendo0plr 36813 tendoipl 36818 tendoid0 36846 tendo0mul 36847 tendo0mulr 36848 tendoex 36996 cdleml5N 37001 erngdvlem1 37009 erngdvlem4 37012 erng0g 37015 erngdvlem1-rN 37017 erngdvlem4-rN 37020 dvh0g 37132 dvhopN 37137 dib1dim 37186 dib1dim2 37189 dibss 37190 diblss 37191 diblsmopel 37192 dicn0 37213 cdlemn4 37219 cdlemn4a 37220 cdlemn6 37223 dihopelvalcpre 37269 dihmeetlem4preN 37327 dihatlat 37355 dihatexv 37359 |
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