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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoco2 | Structured version Visualization version GIF version |
Description: Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.) |
Ref | Expression |
---|---|
tendof.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendof.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendof.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendoco2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑈‘(𝐹 ∘ 𝐺)) ∘ (𝑉‘(𝐹 ∘ 𝐺))) = (((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∘ ((𝑈‘𝐺) ∘ (𝑉‘𝐺)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1196 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐾 ∈ HL) | |
2 | simp1r 1197 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑊 ∈ 𝐻) | |
3 | simp2l 1198 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑈 ∈ 𝐸) | |
4 | simp3l 1200 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐹 ∈ 𝑇) | |
5 | simp3r 1201 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇) | |
6 | tendof.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | tendof.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | tendof.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
9 | 6, 7, 8 | tendovalco 40748 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑈 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘(𝐹 ∘ 𝐺)) = ((𝑈‘𝐹) ∘ (𝑈‘𝐺))) |
10 | 1, 2, 3, 4, 5, 9 | syl32anc 1377 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘(𝐹 ∘ 𝐺)) = ((𝑈‘𝐹) ∘ (𝑈‘𝐺))) |
11 | simp2r 1199 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑉 ∈ 𝐸) | |
12 | 6, 7, 8 | tendovalco 40748 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑉‘(𝐹 ∘ 𝐺)) = ((𝑉‘𝐹) ∘ (𝑉‘𝐺))) |
13 | 1, 2, 11, 4, 5, 12 | syl32anc 1377 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑉‘(𝐹 ∘ 𝐺)) = ((𝑉‘𝐹) ∘ (𝑉‘𝐺))) |
14 | 10, 13 | coeq12d 5878 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑈‘(𝐹 ∘ 𝐺)) ∘ (𝑉‘(𝐹 ∘ 𝐺))) = (((𝑈‘𝐹) ∘ (𝑈‘𝐺)) ∘ ((𝑉‘𝐹) ∘ (𝑉‘𝐺)))) |
15 | simp1 1135 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | 6, 7, 8 | tendocl 40750 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) ∈ 𝑇) |
17 | 15, 3, 5, 16 | syl3anc 1370 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘𝐺) ∈ 𝑇) |
18 | 6, 7, 8 | tendocl 40750 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑉‘𝐹) ∈ 𝑇) |
19 | 15, 11, 4, 18 | syl3anc 1370 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑉‘𝐹) ∈ 𝑇) |
20 | 6, 7 | ltrnco4 40722 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈‘𝐺) ∈ 𝑇 ∧ (𝑉‘𝐹) ∈ 𝑇) → (((𝑈‘𝐹) ∘ (𝑈‘𝐺)) ∘ ((𝑉‘𝐹) ∘ (𝑉‘𝐺))) = (((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∘ ((𝑈‘𝐺) ∘ (𝑉‘𝐺)))) |
21 | 15, 17, 19, 20 | syl3anc 1370 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (((𝑈‘𝐹) ∘ (𝑈‘𝐺)) ∘ ((𝑉‘𝐹) ∘ (𝑉‘𝐺))) = (((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∘ ((𝑈‘𝐺) ∘ (𝑉‘𝐺)))) |
22 | 14, 21 | eqtrd 2775 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑈‘(𝐹 ∘ 𝐺)) ∘ (𝑉‘(𝐹 ∘ 𝐺))) = (((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∘ ((𝑈‘𝐺) ∘ (𝑉‘𝐺)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∘ ccom 5693 ‘cfv 6563 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 TEndoctendo 40735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-undef 8297 df-map 8867 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tendo 40738 |
This theorem is referenced by: tendoplco2 40762 |
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