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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendotp | Structured version Visualization version GIF version |
Description: Trace-preserving property of 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) |
Ref | Expression |
---|---|
tendotp | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | tendoset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendoset.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | tendoset.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
5 | tendoset.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | istendo 40743 | . . 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)))) |
7 | 2fveq3 6912 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘(𝑆‘𝑓)) = (𝑅‘(𝑆‘𝐹))) | |
8 | fveq2 6907 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹)) | |
9 | 7, 8 | breq12d 5161 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) ↔ (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))) |
10 | 9 | rspccv 3619 | . . . 4 ⊢ (∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))) |
11 | 10 | 3ad2ant3 1134 | . . 3 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))) |
12 | 6, 11 | biimtrdi 253 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))) |
13 | 12 | 3imp 1110 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 lecple 17305 LHypclh 39967 LTrncltrn 40084 trLctrl 40141 TEndoctendo 40735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-tendo 40738 |
This theorem is referenced by: tendococl 40755 tendoid 40756 tendopltp 40763 tendoicl 40779 cdlemi1 40801 tendotr 40813 cdleml1N 40959 dva1dim 40968 dialss 41029 diblss 41153 |
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