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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0tp | Structured version Visualization version GIF version |
Description: Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.) |
Ref | Expression |
---|---|
tendo0.b | ⊢ 𝐵 = (Base‘𝐾) |
tendo0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendo0.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendo0.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
tendo0.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
tendo0tp.l | ⊢ ≤ = (le‘𝐾) |
tendo0tp.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendo0tp | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑂‘𝐹)) ≤ (𝑅‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendo0.o | . . . . . 6 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
2 | tendo0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
3 | 1, 2 | tendo02 36855 | . . . . 5 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
4 | 3 | adantl 475 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
5 | 4 | fveq2d 6437 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑂‘𝐹)) = (𝑅‘( I ↾ 𝐵))) |
6 | eqid 2825 | . . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | tendo0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | tendo0tp.r | . . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
9 | 2, 6, 7, 8 | trlid0 36244 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾)) |
10 | 9 | adantr 474 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾)) |
11 | 5, 10 | eqtrd 2861 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑂‘𝐹)) = (0.‘𝐾)) |
12 | hlop 35430 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
13 | 12 | ad2antrr 717 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ OP) |
14 | tendo0.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
15 | 2, 7, 14, 8 | trlcl 36232 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵) |
16 | tendo0tp.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
17 | 2, 16, 6 | op0le 35254 | . . 3 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐹) ∈ 𝐵) → (0.‘𝐾) ≤ (𝑅‘𝐹)) |
18 | 13, 15, 17 | syl2anc 579 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (0.‘𝐾) ≤ (𝑅‘𝐹)) |
19 | 11, 18 | eqbrtrd 4895 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑂‘𝐹)) ≤ (𝑅‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 class class class wbr 4873 ↦ cmpt 4952 I cid 5249 ↾ cres 5344 ‘cfv 6123 Basecbs 16222 lecple 16312 0.cp0 17390 OPcops 35240 HLchlt 35418 LHypclh 36052 LTrncltrn 36169 trLctrl 36226 TEndoctendo 36820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-oposet 35244 df-ol 35246 df-oml 35247 df-covers 35334 df-ats 35335 df-atl 35366 df-cvlat 35390 df-hlat 35419 df-lhyp 36056 df-laut 36057 df-ldil 36172 df-ltrn 36173 df-trl 36227 |
This theorem is referenced by: tendo0cl 36858 |
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