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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia0 | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.) |
Ref | Expression |
---|---|
dia0.b | ⊢ 𝐵 = (Base‘𝐾) |
dia0.z | ⊢ 0 = (0.‘𝐾) |
dia0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia0.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dia0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {( I ↾ 𝐵)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | hlatl 37060 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
3 | dia0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dia0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | atl0cl 37003 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝐾 ∈ HL → 0 ∈ 𝐵) |
7 | 6 | adantr 484 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐵) |
8 | dia0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | 3, 8 | lhpbase 37698 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
10 | eqid 2736 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | 3, 10, 4 | atl0le 37004 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑊 ∈ 𝐵) → 0 (le‘𝐾)𝑊) |
12 | 2, 9, 11 | syl2an 599 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
13 | eqid 2736 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
14 | eqid 2736 | . . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
15 | dia0.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
16 | 3, 10, 8, 13, 14, 15 | diaval 38732 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( 0 ∈ 𝐵 ∧ 0 (le‘𝐾)𝑊)) → (𝐼‘ 0 ) = {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 }) |
17 | 1, 7, 12, 16 | syl12anc 837 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 }) |
18 | 2 | ad2antrr 726 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ AtLat) |
19 | 3, 8, 13, 14 | trlcl 37864 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐵) |
20 | 3, 10, 4 | atlle0 37005 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐵) → ((((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 ↔ (((trL‘𝐾)‘𝑊)‘𝑓) = 0 )) |
21 | 18, 19, 20 | syl2anc 587 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 ↔ (((trL‘𝐾)‘𝑊)‘𝑓) = 0 )) |
22 | 3, 4, 8, 13, 14 | trlid0b 37878 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 = ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) = 0 )) |
23 | 21, 22 | bitr4d 285 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 ↔ 𝑓 = ( I ↾ 𝐵))) |
24 | 23 | rabbidva 3378 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 } = {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ 𝑓 = ( I ↾ 𝐵)}) |
25 | 3, 8, 13 | idltrn 37850 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
26 | rabsn 4623 | . . 3 ⊢ (( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊) → {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ 𝑓 = ( I ↾ 𝐵)} = {( I ↾ 𝐵)}) | |
27 | 25, 26 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ 𝑓 = ( I ↾ 𝐵)} = {( I ↾ 𝐵)}) |
28 | 17, 24, 27 | 3eqtrd 2775 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {( I ↾ 𝐵)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {crab 3055 {csn 4527 class class class wbr 5039 I cid 5439 ↾ cres 5538 ‘cfv 6358 Basecbs 16666 lecple 16756 0.cp0 17883 AtLatcal 36964 HLchlt 37050 LHypclh 37684 LTrncltrn 37801 trLctrl 37858 DIsoAcdia 38728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 df-disoa 38729 |
This theorem is referenced by: dib0 38864 |
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