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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo1mulr | Structured version Visualization version GIF version |
Description: Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.) |
Ref | Expression |
---|---|
tendof.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendof.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendof.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendo1mulr | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendof.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | tendof.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | tendof.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tendof 37898 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → 𝑈:𝑇⟶𝑇) |
5 | fcoi1 6551 | . 2 ⊢ (𝑈:𝑇⟶𝑇 → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) | |
6 | 4, 5 | syl 17 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 I cid 5458 ↾ cres 5556 ∘ ccom 5558 ⟶wf 6350 ‘cfv 6354 HLchlt 36485 LHypclh 37119 LTrncltrn 37236 TEndoctendo 37887 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-tendo 37890 |
This theorem is referenced by: erng1lem 38122 erngdvlem3 38125 erngdvlem3-rN 38133 erngdvlem4-rN 38134 dvhopN 38251 diclspsn 38329 dih1dimatlem0 38463 |
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