Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme7a Structured version   Visualization version   GIF version

Theorem cdleme7a 39846
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 39851 and cdleme7 39852. (Contributed by NM, 7-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l = (le‘𝐾)
cdleme4.j = (join‘𝐾)
cdleme4.m = (meet‘𝐾)
cdleme4.a 𝐴 = (Atoms‘𝐾)
cdleme4.h 𝐻 = (LHyp‘𝐾)
cdleme4.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme4.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme4.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
cdleme7.v 𝑉 = ((𝑅 𝑆) 𝑊)
Assertion
Ref Expression
cdleme7a 𝐺 = ((𝑃 𝑄) (𝐹 𝑉))

Proof of Theorem cdleme7a
StepHypRef Expression
1 cdleme4.g . 2 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
2 cdleme7.v . . . 4 𝑉 = ((𝑅 𝑆) 𝑊)
32oveq2i 7430 . . 3 (𝐹 𝑉) = (𝐹 ((𝑅 𝑆) 𝑊))
43oveq2i 7430 . 2 ((𝑃 𝑄) (𝐹 𝑉)) = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
51, 4eqtr4i 2756 1 𝐺 = ((𝑃 𝑄) (𝐹 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cfv 6549  (class class class)co 7419  lecple 17243  joincjn 18306  meetcmee 18307  Atomscatm 38865  LHypclh 39587
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-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  cdleme7d  39849  cdleme17a  39889
  Copyright terms: Public domain W3C validator