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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo1mul | 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 |
---|---|
tendo1mul | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑈) = 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 40230 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → 𝑈:𝑇⟶𝑇) |
5 | fcoi2 6766 | . 2 ⊢ (𝑈:𝑇⟶𝑇 → (( I ↾ 𝑇) ∘ 𝑈) = 𝑈) | |
6 | 4, 5 | syl 17 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑈) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 I cid 5569 ↾ cres 5674 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 HLchlt 38816 LHypclh 39451 LTrncltrn 39568 TEndoctendo 40219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-tendo 40222 |
This theorem is referenced by: erng1lem 40454 erngdvlem3 40457 erngdvlem3-rN 40465 erngdvlem4-rN 40466 |
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