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

 Description: A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diadmle.h 𝐻 = (LHyp‘𝐾)
diadmle.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diadmleN (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 𝑊)

Proof of Theorem diadmleN
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 diadmle.l . . 3 = (le‘𝐾)
3 diadmle.h . . 3 𝐻 = (LHyp‘𝐾)
4 diadmle.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diaeldm 35343 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Base‘𝐾) ∧ 𝑋 𝑊)))
65simplbda 652 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  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-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-pr 4833 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-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-disoa 35336 This theorem is referenced by:  diaocN  35432  doca2N  35433  djajN  35444
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