Metamath Proof Explorer

Theorem cdleme32sn1awN

Description: Show that [_ R / s ]_ N is an atom not under W when R .<_ ( P .\/ Q ) . (Contributed by NM, 6-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme32.b 
|- B = ( Base ` K )
cdleme32.l 
|- .<_ = ( le ` K )
cdleme32.j 
|- .\/ = ( join ` K )
cdleme32.m 
|- ./\ = ( meet ` K )
cdleme32.a 
|- A = ( Atoms ` K )
cdleme32.h 
|- H = ( LHyp ` K )
cdleme32.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme32.c 
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme32.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme32.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme32.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
cdleme32.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdleme32a1.y 
|- Y = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) )
cdleme32a1.z 
|- Z = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = Y ) )
Assertion cdleme32sn1awN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( [_ R / s ]_ N e. A /\ -. [_ R / s ]_ N .<_ W ) )

Proof

Step Hyp Ref Expression
1 cdleme32.b 
 |-  B = ( Base ` K )
2 cdleme32.l 
 |-  .<_ = ( le ` K )
3 cdleme32.j 
 |-  .\/ = ( join ` K )
4 cdleme32.m 
 |-  ./\ = ( meet ` K )
5 cdleme32.a 
 |-  A = ( Atoms ` K )
6 cdleme32.h 
 |-  H = ( LHyp ` K )
7 cdleme32.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme32.c 
 |-  C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme32.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10 cdleme32.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdleme32.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
12 cdleme32.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
13 cdleme32a1.y 
 |-  Y = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) )
14 cdleme32a1.z 
 |-  Z = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = Y ) )
15 1 2 3 4 5 6 7 9 10 11 12 13 14 cdlemefs32sn1aw 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( [_ R / s ]_ N e. A /\ -. [_ R / s ]_ N .<_ W ) )