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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1N | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dia1.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dia1N | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | dia1.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | lhpbase 39699 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
5 | 4 | adantl 480 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ (Base‘𝐾)) |
6 | hllat 39063 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
7 | eqid 2726 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 2, 7 | latref 18468 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ (Base‘𝐾)) → 𝑊(le‘𝐾)𝑊) |
9 | 6, 4, 8 | syl2an 594 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊(le‘𝐾)𝑊) |
10 | dia1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
11 | eqid 2726 | . . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
12 | dia1.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
13 | 2, 7, 3, 10, 11, 12 | diaval 40733 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊(le‘𝐾)𝑊)) → (𝐼‘𝑊) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊}) |
14 | 1, 5, 9, 13 | syl12anc 835 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊}) |
15 | 7, 3, 10, 11 | trlle 39885 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊) |
16 | 15 | ralrimiva 3136 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊) |
17 | rabid2 3453 | . . 3 ⊢ (𝑇 = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊} ↔ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊) | |
18 | 16, 17 | sylibr 233 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊}) |
19 | 14, 18 | eqtr4d 2769 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 {crab 3419 class class class wbr 5155 ‘cfv 6556 Basecbs 17215 lecple 17275 Latclat 18458 HLchlt 39050 LHypclh 39685 LTrncltrn 39802 trLctrl 39859 DIsoAcdia 40729 |
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 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8859 df-proset 18322 df-poset 18340 df-plt 18357 df-lub 18373 df-glb 18374 df-join 18375 df-meet 18376 df-p0 18452 df-p1 18453 df-lat 18459 df-oposet 38876 df-ol 38878 df-oml 38879 df-covers 38966 df-ats 38967 df-atl 38998 df-cvlat 39022 df-hlat 39051 df-lhyp 39689 df-laut 39690 df-ldil 39805 df-ltrn 39806 df-trl 39860 df-disoa 40730 |
This theorem is referenced by: dia1elN 40755 |
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