Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemd4.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemd4.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemd4.a |
|- A = ( Atoms ` K ) |
4 |
|
cdlemd4.h |
|- H = ( LHyp ` K ) |
5 |
|
cdlemd4.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
fveq2 |
|- ( R = P -> ( F ` R ) = ( F ` P ) ) |
7 |
|
fveq2 |
|- ( R = P -> ( G ` R ) = ( G ` P ) ) |
8 |
6 7
|
eqeq12d |
|- ( R = P -> ( ( F ` R ) = ( G ` R ) <-> ( F ` P ) = ( G ` P ) ) ) |
9 |
|
simpll1 |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) ) |
10 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) ) |
11 |
10
|
adantr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> ( P e. A /\ -. P .<_ W ) ) |
12 |
|
simpl22 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
13 |
12
|
adantr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> ( Q e. A /\ -. Q .<_ W ) ) |
14 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) -> P =/= Q ) |
15 |
14
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> P =/= Q ) |
16 |
|
simplr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> R .<_ ( P .\/ Q ) ) |
17 |
|
simpr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> R =/= P ) |
18 |
15 16 17
|
3jca |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) |
19 |
|
simpll3 |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) |
20 |
1 2 3 4 5
|
cdlemd4 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) -> ( F ` R ) = ( G ` R ) ) |
21 |
9 11 13 18 19 20
|
syl131anc |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) /\ R =/= P ) -> ( F ` R ) = ( G ` R ) ) |
22 |
|
simpl3l |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) -> ( F ` P ) = ( G ` P ) ) |
23 |
8 21 22
|
pm2.61ne |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ R .<_ ( P .\/ Q ) ) -> ( F ` R ) = ( G ` R ) ) |
24 |
|
simpl1 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) ) |
25 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) ) |
26 |
|
simpl22 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
27 |
|
simpl23 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= Q ) |
28 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( P .\/ Q ) ) |
29 |
27 28
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) |
30 |
|
simpl3 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) |
31 |
1 2 3 4 5
|
cdlemd2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) -> ( F ` R ) = ( G ` R ) ) |
32 |
24 25 26 29 30 31
|
syl131anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F ` R ) = ( G ` R ) ) |
33 |
23 32
|
pm2.61dan |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) -> ( F ` R ) = ( G ` R ) ) |