Theorem cdleme3d 30965

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30970 and cdleme3 30971. (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 6084	. . 3 |- ( Q .\/ V ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) )
4	3	oveq2i 6084	. 2 |- ( ( R .\/ U ) ./\ ( Q .\/ V ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
5	1, 4	eqtr4i 2458	1 |- F = ( ( R .\/ U ) ./\ ( Q .\/ V ) )

Syntax hints:   = wceq 1652   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   LHypclh 30718

This theorem is referenced by:  cdleme3g  30968  cdleme3h  30969  cdleme9  30987

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076