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Metamath Proof Explorer

Theorem cdlemd5

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 30-May-2012)

Ref Expression
Hypotheses cdlemd4.l 
|- .<_ = ( le ` K )
cdlemd4.j 
|- .\/ = ( join ` K )
cdlemd4.a 
|- A = ( Atoms ` K )
cdlemd4.h 
|- H = ( LHyp ` K )
cdlemd4.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemd5 
|- ( ( ( ( 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 ) )

Proof

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 ) )