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Theorem cdleme3d 36306
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 36311 and cdleme3 36312. (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 6916 . . 3 (𝑄 𝑉) = (𝑄 ((𝑃 𝑅) 𝑊))
43oveq2i 6916 . 2 ((𝑅 𝑈) (𝑄 𝑉)) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
51, 4eqtr4i 2852 1 𝐹 = ((𝑅 𝑈) (𝑄 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  cfv 6123  (class class class)co 6905  lecple 16312  joincjn 17297  meetcmee 17298  Atomscatm 35338  LHypclh 36059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908
This theorem is referenced by:  cdleme3g  36309  cdleme3h  36310  cdleme9  36328
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