| 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 40765 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:𝑇⟶𝑇) |
| 5 | 4 | 3adant3 1133 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝑆:𝑇⟶𝑇) |
| 6 | simp3 1139 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇) | |
| 7 | 5, 6 | ffvelcdmd 7105 | 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 1087 = wceq 1540 ∈ wcel 2108 ⟶wf 6557 ‘cfv 6561 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-tendo 40757 |
| This theorem is referenced by: tendoco2 40770 tendococl 40774 tendoid 40775 tendoplcl2 40780 tendopltp 40782 tendoplcl 40783 tendoplcom 40784 tendodi1 40786 tendodi2 40787 tendo0pl 40793 tendoicl 40798 tendoipl 40799 cdlemi1 40820 cdlemi2 40821 cdlemi 40822 cdlemj2 40824 tendo0mul 40828 tendoconid 40831 tendotr 40832 cdleml1N 40978 cdleml2N 40979 cdleml6 40983 dva1dim 40987 tendospcl 41020 tendocnv 41023 tendospcanN 41025 dvalveclem 41027 dialss 41048 dvhvscacl 41105 dvhlveclem 41110 dib1dim 41167 dib1dim2 41170 diblss 41172 dicssdvh 41188 diclspsn 41196 cdlemn6 41204 dihopelvalcpre 41250 dih1 41288 dihglbcpreN 41302 dih1dimatlem0 41330 dih1dimatlem 41331 |
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