Mathbox for Norm Megill |
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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 1258 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇)) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))) |
4 | 3 | ralrimivva 2954 | . 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 2949 | . 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 35066 | . . 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 1238 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 lecple 15775 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 TEndoctendo 35058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-tendo 35061 |
This theorem is referenced by: tendoidcl 35075 tendococl 35078 tendoplcl 35087 tendo0cl 35096 tendoicl 35102 cdlemk56 35277 |
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