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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendovalco | Structured version Visualization version GIF version |
Description: Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.) |
Ref | Expression |
---|---|
tendof.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendof.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendof.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendovalco | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | tendof.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendof.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | eqid 2798 | . . . . 5 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
5 | tendof.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | istendo 38056 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓)))) |
7 | coeq1 5692 | . . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔)) | |
8 | 7 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑆‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝑔))) |
9 | fveq2 6645 | . . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑆‘𝑓) = (𝑆‘𝐹)) | |
10 | 9 | coeq1d 5696 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔))) |
11 | 8, 10 | eqeq12d 2814 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)))) |
12 | coeq2 5693 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺)) | |
13 | 12 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑆‘(𝐹 ∘ 𝑔)) = (𝑆‘(𝐹 ∘ 𝐺))) |
14 | fveq2 6645 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑆‘𝑔) = (𝑆‘𝐺)) | |
15 | 14 | coeq2d 5697 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))) |
16 | 13, 15 | eqeq12d 2814 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑆‘(𝐹 ∘ 𝑔)) = ((𝑆‘𝐹) ∘ (𝑆‘𝑔)) ↔ (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))) |
17 | 11, 16 | rspc2v 3581 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))) |
18 | 17 | com12 32 | . . . . 5 ⊢ (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))) |
19 | 18 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓)) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))) |
20 | 6, 19 | syl6bi 256 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))))) |
21 | 20 | 3impia 1114 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) → ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))) |
22 | 21 | imp 410 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 lecple 16564 LHypclh 37280 LTrncltrn 37397 trLctrl 37454 TEndoctendo 38048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-tendo 38051 |
This theorem is referenced by: tendoco2 38064 tendococl 38068 tendodi1 38080 tendoicl 38092 cdlemi2 38115 tendospdi1 38316 |
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