Theorem cdleme7a 41117

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 41122 and cdleme7 41123. (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

Step	Hyp	Ref	Expression
1		cdleme4.g	. 2 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
2		cdleme7.v	. . . 4 ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
3	2	oveq2i 7425	. . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))
4	3	oveq2i 7425	. 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
5	1, 4	eqtr4i 2786	1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉))

This proof depends on syntax axioms:   = wceq 1570  ‘cfv 6533  (class class class)co 7414  lecple 17350  joincjn 18400  meetcmee 18401  Atomscatm 40137  LHypclh 40858

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417

This theorem is used by:  cdleme7d  41120  cdleme17a  41160