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Theorem cdleme7a 36264
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 36269 and cdleme7 36270. (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 6889 . . 3 (𝐹 𝑉) = (𝐹 ((𝑅 𝑆) 𝑊))
43oveq2i 6889 . 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 1653  cfv 6101  (class class class)co 6878  lecple 16274  joincjn 17259  meetcmee 17260  Atomscatm 35284  LHypclh 36005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881
This theorem is referenced by:  cdleme7d  36267  cdleme17a  36307
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