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Theorem cdleme7a 37558
 Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 37563 and cdleme7 37564. (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 7147 . . 3 (𝐹 𝑉) = (𝐹 ((𝑅 𝑆) 𝑊))
43oveq2i 7147 . 2 ((𝑃 𝑄) (𝐹 𝑉)) = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
51, 4eqtr4i 2824 1 𝐺 = ((𝑃 𝑄) (𝐹 𝑉))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ‘cfv 6325  (class class class)co 7136  lecple 16567  joincjn 17549  meetcmee 17550  Atomscatm 36578  LHypclh 37299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-ov 7139 This theorem is referenced by:  cdleme7d  37561  cdleme17a  37601
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