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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0co2 | Structured version Visualization version GIF version | ||
| Description: The additive identity trace-preserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 41022? (Contributed by NM, 11-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 |
|---|---|
| tendo0co2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑂‘(𝐹 ∘ 𝐺)) = ((𝑂‘𝐹) ∘ (𝑂‘𝐺))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tendo0.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | tendo0.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 3 | 1, 2 | ltrnco 40721 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇) |
| 4 | tendo0.o | . . . 4 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
| 5 | tendo0.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 6 | 4, 5 | tendo02 40789 | . . 3 ⊢ ((𝐹 ∘ 𝐺) ∈ 𝑇 → (𝑂‘(𝐹 ∘ 𝐺)) = ( I ↾ 𝐵)) |
| 7 | 3, 6 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑂‘(𝐹 ∘ 𝐺)) = ( I ↾ 𝐵)) |
| 8 | 4, 5 | tendo02 40789 | . . . . 5 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
| 9 | 8 | 3ad2ant2 1135 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
| 10 | 4, 5 | tendo02 40789 | . . . . 5 ⊢ (𝐺 ∈ 𝑇 → (𝑂‘𝐺) = ( I ↾ 𝐵)) |
| 11 | 10 | 3ad2ant3 1136 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑂‘𝐺) = ( I ↾ 𝐵)) |
| 12 | 9, 11 | coeq12d 5875 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑂‘𝐹) ∘ (𝑂‘𝐺)) = (( I ↾ 𝐵) ∘ ( I ↾ 𝐵))) |
| 13 | f1oi 6886 | . . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
| 14 | f1of 6848 | . . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
| 15 | fcoi1 6782 | . . . 4 ⊢ (( I ↾ 𝐵):𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ ( I ↾ 𝐵)) = ( I ↾ 𝐵)) | |
| 16 | 13, 14, 15 | mp2b 10 | . . 3 ⊢ (( I ↾ 𝐵) ∘ ( I ↾ 𝐵)) = ( I ↾ 𝐵) |
| 17 | 12, 16 | eqtr2di 2794 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ( I ↾ 𝐵) = ((𝑂‘𝐹) ∘ (𝑂‘𝐺))) |
| 18 | 7, 17 | eqtrd 2777 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑂‘(𝐹 ∘ 𝐺)) = ((𝑂‘𝐹) ∘ (𝑂‘𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 I cid 5577 ↾ cres 5687 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 Basecbs 17247 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 |
| This theorem is referenced by: tendo0cl 40792 |
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