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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoipl2 | Structured version Visualization version GIF version |
Description: Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.) |
Ref | Expression |
---|---|
tendoicl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendoicl.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendoicl.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
tendoicl.i | ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) |
tendoi.b | ⊢ 𝐵 = (Base‘𝐾) |
tendoi.p | ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
tendoi.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
tendoipl2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoicl.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | tendoicl.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | tendoicl.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | tendoicl.i | . . . 4 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) | |
5 | 1, 2, 3, 4 | tendoicl 39656 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝐼‘𝑆) ∈ 𝐸) |
6 | tendoi.p | . . . 4 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
7 | 1, 2, 3, 6 | tendoplcom 39642 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ (𝐼‘𝑆) ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = ((𝐼‘𝑆)𝑃𝑆)) |
8 | 5, 7 | mpd3an3 1463 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = ((𝐼‘𝑆)𝑃𝑆)) |
9 | tendoi.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
10 | tendoi.o | . . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
11 | 1, 2, 3, 4, 9, 6, 10 | tendoipl 39657 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → ((𝐼‘𝑆)𝑃𝑆) = 𝑂) |
12 | 8, 11 | eqtrd 2773 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5231 I cid 5573 ◡ccnv 5675 ↾ cres 5678 ∘ ccom 5680 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 Basecbs 17141 HLchlt 38209 LHypclh 38844 LTrncltrn 38961 TEndoctendo 39612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-riotaBAD 37812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-undef 8255 df-map 8819 df-proset 18245 df-poset 18263 df-plt 18280 df-lub 18296 df-glb 18297 df-join 18298 df-meet 18299 df-p0 18375 df-p1 18376 df-lat 18382 df-clat 18449 df-oposet 38035 df-ol 38037 df-oml 38038 df-covers 38125 df-ats 38126 df-atl 38157 df-cvlat 38181 df-hlat 38210 df-llines 38358 df-lplanes 38359 df-lvols 38360 df-lines 38361 df-psubsp 38363 df-pmap 38364 df-padd 38656 df-lhyp 38848 df-laut 38849 df-ldil 38964 df-ltrn 38965 df-trl 39019 df-tendo 39615 |
This theorem is referenced by: dihjatcclem4 40281 |
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