Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dia1N Structured version   Visualization version   GIF version

Theorem dia1N 40427
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.)
Hypotheses
Ref Expression
dia1.h 𝐻 = (LHypβ€˜πΎ)
dia1.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dia1.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dia1N ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = 𝑇)

Proof of Theorem dia1N
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 eqid 2724 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 dia1.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3lhpbase 39372 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
54adantl 481 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š ∈ (Baseβ€˜πΎ))
6 hllat 38736 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
7 eqid 2724 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
82, 7latref 18402 . . . 4 ((𝐾 ∈ Lat ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ π‘Š(leβ€˜πΎ)π‘Š)
96, 4, 8syl2an 595 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š(leβ€˜πΎ)π‘Š)
10 dia1.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
11 eqid 2724 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
12 dia1.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
132, 7, 3, 10, 11, 12diaval 40406 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Š ∈ (Baseβ€˜πΎ) ∧ π‘Š(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜π‘Š) = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š})
141, 5, 9, 13syl12anc 834 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š})
157, 3, 10, 11trlle 39558 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
1615ralrimiva 3138 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆ€π‘“ ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
17 rabid2 3456 . . 3 (𝑇 = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š} ↔ βˆ€π‘“ ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
1816, 17sylibr 233 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑇 = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š})
1914, 18eqtr4d 2767 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  {crab 3424   class class class wbr 5139  β€˜cfv 6534  Basecbs 17149  lecple 17209  Latclat 18392  HLchlt 38723  LHypclh 39358  LTrncltrn 39475  trLctrl 39532  DIsoAcdia 40402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-p1 18387  df-lat 18393  df-oposet 38549  df-ol 38551  df-oml 38552  df-covers 38639  df-ats 38640  df-atl 38671  df-cvlat 38695  df-hlat 38724  df-lhyp 39362  df-laut 39363  df-ldil 39478  df-ltrn 39479  df-trl 39533  df-disoa 40403
This theorem is referenced by:  dia1elN  40428
  Copyright terms: Public domain W3C validator