Metamath Proof Explorer


Theorem cdleme43frv1snN

Description: Value of [_ R / s ]_ N when -. R .<_ ( P .\/ Q ) . (Contributed by NM, 30-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemefr27.b
|- B = ( Base ` K )
cdlemefr27.l
|- .<_ = ( le ` K )
cdlemefr27.j
|- .\/ = ( join ` K )
cdlemefr27.m
|- ./\ = ( meet ` K )
cdlemefr27.a
|- A = ( Atoms ` K )
cdlemefr27.h
|- H = ( LHyp ` K )
cdlemefr27.u
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefr27.c
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdlemefr27.n
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdleme43fr.x
|- X = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdleme43frv1snN
|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = X )

Proof

Step Hyp Ref Expression
1 cdlemefr27.b
 |-  B = ( Base ` K )
2 cdlemefr27.l
 |-  .<_ = ( le ` K )
3 cdlemefr27.j
 |-  .\/ = ( join ` K )
4 cdlemefr27.m
 |-  ./\ = ( meet ` K )
5 cdlemefr27.a
 |-  A = ( Atoms ` K )
6 cdlemefr27.h
 |-  H = ( LHyp ` K )
7 cdlemefr27.u
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefr27.c
 |-  C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdlemefr27.n
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
10 cdleme43fr.x
 |-  X = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
11 8 9 10 cdleme31sn2
 |-  ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = X )