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Theorem cdleme7a 35010
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 35015 and cdleme7 35016. (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 6615 . . 3 (𝐹 𝑉) = (𝐹 ((𝑅 𝑆) 𝑊))
43oveq2i 6615 . 2 ((𝑃 𝑄) (𝐹 𝑉)) = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
51, 4eqtr4i 2646 1 𝐺 = ((𝑃 𝑄) (𝐹 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cfv 5847  (class class class)co 6604  lecple 15869  joincjn 16865  meetcmee 16866  Atomscatm 34030  LHypclh 34750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  cdleme7d  35013  cdleme17a  35053
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