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 ) ) |
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 ) ) |