Metamath Proof Explorer

Theorem cdleme43aN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v_1). (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme43.b 
|- B = ( Base ` K )
cdleme43.l 
|- .<_ = ( le ` K )
cdleme43.j 
|- .\/ = ( join ` K )
cdleme43.m 
|- ./\ = ( meet ` K )
cdleme43.a 
|- A = ( Atoms ` K )
cdleme43.h 
|- H = ( LHyp ` K )
cdleme43.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme43.x 
|- X = ( ( Q .\/ P ) ./\ W )
cdleme43.c 
|- C = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme43.f 
|- Z = ( ( P .\/ Q ) ./\ ( C .\/ ( ( R .\/ S ) ./\ W ) ) )
cdleme43.d 
|- D = ( ( S .\/ X ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )
cdleme43.g 
|- G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
cdleme43.e 
|- E = ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) )
cdleme43.v 
|- V = ( ( Z .\/ S ) ./\ W )
cdleme43.y 
|- Y = ( ( R .\/ D ) ./\ W )
Assertion cdleme43aN 
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> G = ( ( P .\/ Q ) ./\ ( D .\/ V ) ) )

Proof

Step Hyp Ref Expression
1 cdleme43.b 
 |-  B = ( Base ` K )
2 cdleme43.l 
 |-  .<_ = ( le ` K )
3 cdleme43.j 
 |-  .\/ = ( join ` K )
4 cdleme43.m 
 |-  ./\ = ( meet ` K )
5 cdleme43.a 
 |-  A = ( Atoms ` K )
6 cdleme43.h 
 |-  H = ( LHyp ` K )
7 cdleme43.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme43.x 
 |-  X = ( ( Q .\/ P ) ./\ W )
9 cdleme43.c 
 |-  C = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
10 cdleme43.f 
 |-  Z = ( ( P .\/ Q ) ./\ ( C .\/ ( ( R .\/ S ) ./\ W ) ) )
11 cdleme43.d 
 |-  D = ( ( S .\/ X ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )
12 cdleme43.g 
 |-  G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
13 cdleme43.e 
 |-  E = ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) )
14 cdleme43.v 
 |-  V = ( ( Z .\/ S ) ./\ W )
15 cdleme43.y 
 |-  Y = ( ( R .\/ D ) ./\ W )
16 3 5 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
17 14 oveq2i 
 |-  ( D .\/ V ) = ( D .\/ ( ( Z .\/ S ) ./\ W ) )
18 17 a1i 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( D .\/ V ) = ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
19 16 18 oveq12d 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P .\/ Q ) ./\ ( D .\/ V ) ) = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) ) )
20 12 19 eqtr4id 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> G = ( ( P .\/ Q ) ./\ ( D .\/ V ) ) )