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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendocl | Structured version Visualization version GIF version |
Description: Closure of 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 |
---|---|
tendocl | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendof.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | tendof.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | tendof.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tendof 40745 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:𝑇⟶𝑇) |
5 | 4 | 3adant3 1131 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝑆:𝑇⟶𝑇) |
6 | simp3 1137 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇) | |
7 | 5, 6 | ffvelcdmd 7104 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ⟶wf 6558 ‘cfv 6562 LHypclh 39966 LTrncltrn 40083 TEndoctendo 40734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-tendo 40737 |
This theorem is referenced by: tendoco2 40750 tendococl 40754 tendoid 40755 tendoplcl2 40760 tendopltp 40762 tendoplcl 40763 tendoplcom 40764 tendodi1 40766 tendodi2 40767 tendo0pl 40773 tendoicl 40778 tendoipl 40779 cdlemi1 40800 cdlemi2 40801 cdlemi 40802 cdlemj2 40804 tendo0mul 40808 tendoconid 40811 tendotr 40812 cdleml1N 40958 cdleml2N 40959 cdleml6 40963 dva1dim 40967 tendospcl 41000 tendocnv 41003 tendospcanN 41005 dvalveclem 41007 dialss 41028 dvhvscacl 41085 dvhlveclem 41090 dib1dim 41147 dib1dim2 41150 diblss 41152 dicssdvh 41168 diclspsn 41176 cdlemn6 41184 dihopelvalcpre 41230 dih1 41268 dihglbcpreN 41282 dih1dimatlem0 41310 dih1dimatlem 41311 |
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