![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2772 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | dia1.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | lhpbase 36608 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
5 | 4 | adantl 474 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ (Base‘𝐾)) |
6 | hllat 35973 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
7 | eqid 2772 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 2, 7 | latref 17533 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ (Base‘𝐾)) → 𝑊(le‘𝐾)𝑊) |
9 | 6, 4, 8 | syl2an 586 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊(le‘𝐾)𝑊) |
10 | dia1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
11 | eqid 2772 | . . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
12 | dia1.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
13 | 2, 7, 3, 10, 11, 12 | diaval 37642 | . . 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 824 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊}) |
15 | 7, 3, 10, 11 | trlle 36794 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊) |
16 | 15 | ralrimiva 3126 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊) |
17 | rabid2 3314 | . . 3 ⊢ (𝑇 = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊} ↔ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊) | |
18 | 16, 17 | sylibr 226 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓)(le‘𝐾)𝑊}) |
19 | 14, 18 | eqtr4d 2811 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3082 {crab 3086 class class class wbr 4925 ‘cfv 6185 Basecbs 16337 lecple 16426 Latclat 17525 HLchlt 35960 LHypclh 36594 LTrncltrn 36711 trLctrl 36768 DIsoAcdia 37638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-map 8206 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-oposet 35786 df-ol 35788 df-oml 35789 df-covers 35876 df-ats 35877 df-atl 35908 df-cvlat 35932 df-hlat 35961 df-lhyp 36598 df-laut 36599 df-ldil 36714 df-ltrn 36715 df-trl 36769 df-disoa 37639 |
This theorem is referenced by: dia1elN 37664 |
Copyright terms: Public domain | W3C validator |