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| Mirrors > Home > MPE Home > Th. List > Mathboxes > istendod | Structured version Visualization version GIF version | ||
| Description: Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.) | 
| Ref | Expression | 
|---|---|
| tendoset.l | ⊢ ≤ = (le‘𝐾) | 
| tendoset.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| tendoset.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | 
| tendoset.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | 
| tendoset.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | 
| istendod.1 | ⊢ (𝜑 → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻)) | 
| istendod.2 | ⊢ (𝜑 → 𝑆:𝑇⟶𝑇) | 
| istendod.3 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))) | 
| istendod.4 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) | 
| Ref | Expression | 
|---|---|
| istendod | ⊢ (𝜑 → 𝑆 ∈ 𝐸) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | istendod.2 | . 2 ⊢ (𝜑 → 𝑆:𝑇⟶𝑇) | |
| 2 | istendod.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))) | |
| 3 | 2 | 3expb 1120 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇)) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))) | 
| 4 | 3 | ralrimivva 3201 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))) | 
| 5 | istendod.4 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) | |
| 6 | 5 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) | 
| 7 | istendod.1 | . . 3 ⊢ (𝜑 → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻)) | |
| 8 | tendoset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 9 | tendoset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 10 | tendoset.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 11 | tendoset.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
| 12 | tendoset.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
| 13 | 8, 9, 10, 11, 12 | istendo 40763 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)))) | 
| 14 | 7, 13 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)))) | 
| 15 | 1, 4, 6, 14 | mpbir3and 1342 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐸) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 lecple 17305 LHypclh 39987 LTrncltrn 40104 trLctrl 40161 TEndoctendo 40755 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-tendo 40758 | 
| This theorem is referenced by: tendoidcl 40772 tendococl 40775 tendoplcl 40784 tendo0cl 40793 tendoicl 40799 cdlemk56 40974 | 
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