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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 39572 | . . 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 7083 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⟶wf 6536 ‘cfv 6540 LHypclh 38793 LTrncltrn 38910 TEndoctendo 39561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8818 df-tendo 39564 |
This theorem is referenced by: tendoco2 39577 tendococl 39581 tendoid 39582 tendoplcl2 39587 tendopltp 39589 tendoplcl 39590 tendoplcom 39591 tendodi1 39593 tendodi2 39594 tendo0pl 39600 tendoicl 39605 tendoipl 39606 cdlemi1 39627 cdlemi2 39628 cdlemi 39629 cdlemj2 39631 tendo0mul 39635 tendoconid 39638 tendotr 39639 cdleml1N 39785 cdleml2N 39786 cdleml6 39790 dva1dim 39794 tendospcl 39827 tendocnv 39830 tendospcanN 39832 dvalveclem 39834 dialss 39855 dvhvscacl 39912 dvhlveclem 39917 dib1dim 39974 dib1dim2 39977 diblss 39979 dicssdvh 39995 diclspsn 40003 cdlemn6 40011 dihopelvalcpre 40057 dih1 40095 dihglbcpreN 40109 dih1dimatlem0 40137 dih1dimatlem 40138 |
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