Theorem cdleme3d 29571

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29576 and cdleme3 29577. (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 5789	. . 3 |- ( Q .\/ V ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) )
4	3	oveq2i 5789	. 2 |- ( ( R .\/ U ) ./\ ( Q .\/ V ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
5	1, 4	eqtr4i 2279	1 |- F = ( ( R .\/ U ) ./\ ( Q .\/ V ) )

Syntax hints:   = wceq 1619   ` cfv 4659  (class class class)co 5778   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   LHypclh 29324

This theorem is referenced by:  cdleme3g  29574  cdleme3h  29575  cdleme9  29593

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fv 4675  df-ov 5781