Theorem cdleme3d 35518

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 35523 and cdleme3 35524. (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 6661	. . 3 |- ( Q .\/ V ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) )
4	3	oveq2i 6661	. 2 |- ( ( R .\/ U ) ./\ ( Q .\/ V ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
5	1, 4	eqtr4i 2647	1 |- F = ( ( R .\/ U ) ./\ ( Q .\/ V ) )

Syntax hints:   = wceq 1483   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   LHypclh 35270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  cdleme3g  35521  cdleme3h  35522  cdleme9  35540