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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > istendo | Structured version Visualization version GIF version |
Description: The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-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 |
---|---|
istendo | ⊢ ((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)))) |
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 | tendoset 36780 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))}) |
7 | 6 | eleq2d 2864 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ 𝑆 ∈ {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})) |
8 | 3 | fvexi 6425 | . . . . 5 ⊢ 𝑇 ∈ V |
9 | fex 6718 | . . . . 5 ⊢ ((𝑆:𝑇⟶𝑇 ∧ 𝑇 ∈ V) → 𝑆 ∈ V) | |
10 | 8, 9 | mpan2 683 | . . . 4 ⊢ (𝑆:𝑇⟶𝑇 → 𝑆 ∈ V) |
11 | 10 | 3ad2ant1 1164 | . . 3 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) → 𝑆 ∈ V) |
12 | feq1 6237 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑠:𝑇⟶𝑇 ↔ 𝑆:𝑇⟶𝑇)) | |
13 | fveq1 6410 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝑓 ∘ 𝑔))) | |
14 | fveq1 6410 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑓) = (𝑆‘𝑓)) | |
15 | fveq1 6410 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑔) = (𝑆‘𝑔)) | |
16 | 14, 15 | coeq12d 5490 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))) |
17 | 13, 16 | eqeq12d 2814 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))) |
18 | 17 | 2ralbidv 3170 | . . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))) |
19 | 14 | fveq2d 6415 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅‘(𝑠‘𝑓)) = (𝑅‘(𝑆‘𝑓))) |
20 | 19 | breq1d 4853 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓) ↔ (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))) |
21 | 20 | ralbidv 3167 | . . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓) ↔ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))) |
22 | 12, 18, 21 | 3anbi123d 1561 | . . 3 ⊢ (𝑠 = 𝑆 → ((𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)) ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)))) |
23 | 11, 22 | elab3 3550 | . 2 ⊢ (𝑆 ∈ {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))} ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))) |
24 | 7, 23 | syl6bb 279 | 1 ⊢ ((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)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {cab 2785 ∀wral 3089 Vcvv 3385 class class class wbr 4843 ∘ ccom 5316 ⟶wf 6097 ‘cfv 6101 lecple 16274 LHypclh 36005 LTrncltrn 36122 trLctrl 36179 TEndoctendo 36773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-map 8097 df-tendo 36776 |
This theorem is referenced by: tendotp 36782 istendod 36783 tendof 36784 tendovalco 36786 |
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