Theorem cdleme3d 39790

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 39795 and cdleme3 39796. (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

Step	Hyp	Ref	Expression
1		cdleme1.f	. 2 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
2		cdleme3.3	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊)
3	2	oveq2i 7428	. . 3 ⊢ (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))
4	3	oveq2i 7428	. 2 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
5	1, 4	eqtr4i 2756	1 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉))

Syntax hints:   = wceq 1533  ‘cfv 6547  (class class class)co 7417  lecple 17240  joincjn 18303  meetcmee 18304  Atomscatm 38821  LHypclh 39543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6499  df-fv 6555  df-ov 7420

This theorem is referenced by:  cdleme3g  39793  cdleme3h  39794  cdleme9  39812

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »