Theorem cdleme3d 30717

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30722 and cdleme3 30723. (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 6055	. . 3 |- ( Q .\/ V ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) )
4	3	oveq2i 6055	. 2 |- ( ( R .\/ U ) ./\ ( Q .\/ V ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
5	1, 4	eqtr4i 2431	1 |- F = ( ( R .\/ U ) ./\ ( Q .\/ V ) )

Syntax hints:   = wceq 1649   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   LHypclh 30470

This theorem is referenced by:  cdleme3g  30720  cdleme3h  30721  cdleme9  30739

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-ov 6047