Theorem cdleme3d 30420

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30425 and cdleme3 30426. (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 5869	. . 3 |- ( Q .\/ V ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) )
4	3	oveq2i 5869	. 2 |- ( ( R .\/ U ) ./\ ( Q .\/ V ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
5	1, 4	eqtr4i 2306	1 |- F = ( ( R .\/ U ) ./\ ( Q .\/ V ) )

Syntax hints:   = wceq 1623   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173

This theorem is referenced by:  cdleme3g  30423  cdleme3h  30424  cdleme9  30442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861