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Theorem cdlemd4 35488
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
Hypotheses
Ref Expression
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
Ref Expression
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 ) )
Proof of Theorem cdlemd4
Dummy variable s is distinct from all other variables.
StepHypRef Expression
1 simp11l 1172 . . 3 |- ( ( ( ( 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 ) ) ) -> K e. HL )
2 simp11r 1173 . . 3 |- ( ( ( ( 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 ) ) ) -> W e. H )
3 simp21 1094 . . 3 |- ( ( ( ( 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 ) ) ) -> ( P e. A /\ -. P .<_ W ) )
4 simp22 1095 . . 3 |- ( ( ( ( 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 ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
5 simp231 1205 . . 3 |- ( ( ( ( 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 ) ) ) -> P =/= Q )
6 cdlemd4.l . . . 4 |- .<_ = ( le ` K )
7 cdlemd4.j . . . 4 |- .\/ = ( join ` K )
8 cdlemd4.a . . . 4 |- A = ( Atoms ` K )
9 cdlemd4.h . . . 4 |- H = ( LHyp ` K )
106, 7, 8, 9cdlemb2 35327 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. s e. A ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) )
111, 2, 3, 4, 5, 10syl221anc 1337 . 2 |- ( ( ( ( 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 ) ) ) -> E. s e. A ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) )
12 simpl11 1136 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
13 simpl12 1137 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( F e. T /\ G e. T ) )
14 simpl13 1138 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> R e. A )
15 simpl21 1139 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
16 simprl 794 . . . 4 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> s e. A )
17 simprrl 804 . . . 4 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> -. s .<_ W )
1816, 17jca 554 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( s e. A /\ -. s .<_ W ) )
19 hllat 34650 . . . . . . 7 |- ( K e. HL -> K e. Lat )
201, 19syl 17 . . . . . 6 |- ( ( ( ( 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 ) ) ) -> K e. Lat )
2120adantr 481 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> K e. Lat )
22 eqid 2622 . . . . . . 7 |- ( Base ` K ) = ( Base ` K )
2322, 8atbase 34576 . . . . . 6 |- ( s e. A -> s e. ( Base ` K ) )
2423ad2antrl 764 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> s e. ( Base ` K ) )
25 simp21l 1178 . . . . . . 7 |- ( ( ( ( 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 ) ) ) -> P e. A )
2622, 8atbase 34576 . . . . . . 7 |- ( P e. A -> P e. ( Base ` K ) )
2725, 26syl 17 . . . . . 6 |- ( ( ( ( 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 ) ) ) -> P e. ( Base ` K ) )
2827adantr 481 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> P e. ( Base ` K ) )
29 simp22l 1180 . . . . . . 7 |- ( ( ( ( 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 ) ) ) -> Q e. A )
3022, 8atbase 34576 . . . . . . 7 |- ( Q e. A -> Q e. ( Base ` K ) )
3129, 30syl 17 . . . . . 6 |- ( ( ( ( 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 ) ) ) -> Q e. ( Base ` K ) )
3231adantr 481 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> Q e. ( Base ` K ) )
33 simprrr 805 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> -. s .<_ ( P .\/ Q ) )
3422, 6, 7latnlej1l 17069 . . . . . 6 |- ( ( K e. Lat /\ ( s e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. s .<_ ( P .\/ Q ) ) -> s =/= P )
3534necomd 2849 . . . . 5 |- ( ( K e. Lat /\ ( s e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. s .<_ ( P .\/ Q ) ) -> P =/= s )
3621, 24, 28, 32, 33, 35syl131anc 1339 . . . 4 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> P =/= s )
37 simpl22 1140 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
38 simpl23 1141 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) )
396, 7, 8, 9cdlemd3 35487 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ s e. A /\ -. s .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ s ) )
4012, 15, 37, 38, 14, 16, 33, 39syl133anc 1349 . . . 4 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> -. R .<_ ( P .\/ s ) )
4136, 40jca 554 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( P =/= s /\ -. R .<_ ( P .\/ s ) ) )
42 simpl3l 1116 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( F ` P ) = ( G ` P ) )
435adantr 481 . . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> P =/= Q )
4443, 33jca 554 . . . 4 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( P =/= Q /\ -. s .<_ ( P .\/ Q ) ) )
45 simpl3 1066 . . . 4 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) )
46 cdlemd4.t . . . . 5 |- T = ( ( LTrn ` K ) ` W )
476, 7, 8, 9, 46cdlemd2 35486 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ s e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ -. s .<_ ( P .\/ Q ) ) ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` Q ) = ( G ` Q ) ) ) -> ( F ` s ) = ( G ` s ) )
4812, 13, 16, 15, 37, 44, 45, 47syl331anc 1351 . . 3 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( F ` s ) = ( G ` s ) )
496, 7, 8, 9, 46cdlemd2 35486 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( s e. A /\ -. s .<_ W ) /\ ( P =/= s /\ -. R .<_ ( P .\/ s ) ) ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` s ) = ( G ` s ) ) ) -> ( F ` R ) = ( G ` R ) )
5012, 13, 14, 15, 18, 41, 42, 48, 49syl332anc 1357 . 2 |- ( ( ( ( ( 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 ) ) ) /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( F ` R ) = ( G ` R ) )
5111, 50rexlimddv 3035 1 |- ( ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391
This theorem is referenced by:  cdlemd5  35489
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