Mathbox for Norm Megill |
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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 33665 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
3 | dia0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dia0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | atl0cl 33608 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝐾 ∈ HL → 0 ∈ 𝐵) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐵) |
8 | dia0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | 3, 8 | lhpbase 34302 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
10 | eqid 2610 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | 3, 10, 4 | atl0le 33609 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑊 ∈ 𝐵) → 0 (le‘𝐾)𝑊) |
12 | 2, 9, 11 | syl2an 493 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
13 | eqid 2610 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
14 | eqid 2610 | . . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
15 | dia0.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
16 | 3, 10, 8, 13, 14, 15 | diaval 35339 | . . 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 1316 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 }) |
18 | 2 | ad2antrr 758 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ AtLat) |
19 | 3, 8, 13, 14 | trlcl 34469 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐵) |
20 | 3, 10, 4 | atlle0 33610 | . . . . 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 691 | . . . 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 34483 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 = ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) = 0 )) |
23 | 21, 22 | bitr4d 270 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾) 0 ↔ 𝑓 = ( I ↾ 𝐵))) |
24 | 23 | rabbidva 3163 | . 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 34454 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
26 | rabsn 4200 | . . 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 2648 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {( I ↾ 𝐵)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 {csn 4125 class class class wbr 4583 I cid 4948 ↾ cres 5040 ‘cfv 5804 Basecbs 15695 lecple 15775 0.cp0 16860 AtLatcal 33569 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 DIsoAcdia 35335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-disoa 35336 |
This theorem is referenced by: dib0 35471 |
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