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Theorem cdleme3d 34536
 Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 34541 and cdleme3 34542. (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 6560 . . 3 (𝑄 𝑉) = (𝑄 ((𝑃 𝑅) 𝑊))
43oveq2i 6560 . 2 ((𝑅 𝑈) (𝑄 𝑉)) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
51, 4eqtr4i 2635 1 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  LHypclh 34288 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  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-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  cdleme3g  34539  cdleme3h  34540  cdleme9  34558
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