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Theorem cdleme3d 38907
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38912 and cdleme3 38913. (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l = (le‘𝐾)
cdleme1.j = (join‘𝐾)
cdleme1.m = (meet‘𝐾)
cdleme1.a 𝐴 = (Atoms‘𝐾)
cdleme1.h 𝐻 = (LHyp‘𝐾)
cdleme1.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme1.f 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
cdleme3.3 𝑉 = ((𝑃 𝑅) 𝑊)
Assertion
Ref Expression
cdleme3d 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))

Proof of Theorem cdleme3d
StepHypRef Expression
1 cdleme1.f . 2 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
2 cdleme3.3 . . . 4 𝑉 = ((𝑃 𝑅) 𝑊)
32oveq2i 7404 . . 3 (𝑄 𝑉) = (𝑄 ((𝑃 𝑅) 𝑊))
43oveq2i 7404 . 2 ((𝑅 𝑈) (𝑄 𝑉)) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
51, 4eqtr4i 2762 1 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cfv 6532  (class class class)co 7393  lecple 17186  joincjn 18246  meetcmee 18247  Atomscatm 37938  LHypclh 38660
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-iota 6484  df-fv 6540  df-ov 7396
This theorem is referenced by:  cdleme3g  38910  cdleme3h  38911  cdleme9  38929
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