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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 36048 | . . 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 | fveq2 6191 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑆‘𝑓) = (𝑆‘𝐹)) | |
8 | 7 | fveq2d 6195 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘(𝑆‘𝑓)) = (𝑅‘(𝑆‘𝐹))) |
9 | fveq2 6191 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹)) | |
10 | 8, 9 | breq12d 4666 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) ↔ (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))) |
11 | 10 | rspccv 3306 | . . . 4 ⊢ (∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))) |
12 | 11 | 3ad2ant3 1084 | . . 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))) |
13 | 6, 12 | syl6bi 243 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))) |
14 | 13 | 3imp 1256 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 lecple 15948 LHypclh 35270 LTrncltrn 35387 trLctrl 35445 TEndoctendo 36040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-tendo 36043 |
This theorem is referenced by: tendococl 36060 tendoid 36061 tendopltp 36068 tendoicl 36084 cdlemi1 36106 tendotr 36118 cdleml1N 36264 dva1dim 36273 dialss 36335 diblss 36459 |
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