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Theorem cdleme3d 40214
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 40219 and cdleme3 40220. (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 7442 . . 3 (𝑄 𝑉) = (𝑄 ((𝑃 𝑅) 𝑊))
43oveq2i 7442 . 2 ((𝑅 𝑈) (𝑄 𝑉)) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
51, 4eqtr4i 2766 1 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  cdleme3g  40217  cdleme3h  40218  cdleme9  40236
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