Theorem cdlemk20 35545

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be f_i. Our D, C, O, Q, U, V represent their f₁, f₂, k₁, k₂, sigma₁, sigma₂. (Contributed by NM, 5-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk1.b	|- B = ( Base ` K )
cdlemk1.l	|- .<_ = ( le ` K )
cdlemk1.j	|- .\/ = ( join ` K )
cdlemk1.m	|- ./\ = ( meet ` K )
cdlemk1.a	|- A = ( Atoms ` K )
cdlemk1.h	|- H = ( LHyp ` K )
cdlemk1.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r	|- R = ( ( trL ` K ) ` W )
cdlemk1.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o	|- O = ( S ` D )
cdlemk1.u	|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
cdlemk2a.q	|- Q = ( S ` C )

Assertion

Ref	Expression
cdlemk20	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( U ` C ) ` P ) = ( Q ` P ) )

Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   D, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i   ./\ , e   .\/ , e   D, e, j   e, O   P, e   R, e   T, e   e, W   ./\ , j   .<_ , j   .\/ , j   A, j   D, j   j, F   j, H   j, K   j, N   j, O   P, j   R, j   T, j   j, W   e, F, f, i   C, e   f, j, C, i

Allowed substitution hints:   A( e, f)   B( e, f, i, j)   Q( e, f, i, j)   S( e, f, i, j)   U( e, f, i, j)   H( e, f)   K( e, f)   .<_ ( e, f)   N( e)   O( f, i)

Proof of Theorem cdlemk20

Step	Hyp	Ref	Expression
1		simp11 1021	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp23 1026	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( R ` F ) = ( R ` N ) )
3		simp21r 1109	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> C e. T )
4		simp12 1022	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> F e. T )
5		simp13 1023	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> D e. T )
6		simp21l 1108	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> N e. T )
7		simp3r1 1099	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
8		simp3r3 1101	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( R ` C ) =/= ( R ` D ) )
9	8	necomd 2731	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` C ) )
10	7, 9	jca 532	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` C ) ) )
11		simp3l1 1096	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> F =/= ( _I |` B ) )
12		simp3l3 1098	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> C =/= ( _I |` B ) )
13		simp3l2 1097	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> D =/= ( _I |` B ) )
14	11, 12, 13	3jca 1171	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) )
15		simp22 1025	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
16		cdlemk1.b	. . . 4 |- B = ( Base ` K )
17		cdlemk1.l	. . . 4 |- .<_ = ( le ` K )
18		cdlemk1.j	. . . 4 |- .\/ = ( join ` K )
19		cdlemk1.m	. . . 4 |- ./\ = ( meet ` K )
20		cdlemk1.a	. . . 4 |- A = ( Atoms ` K )
21		cdlemk1.h	. . . 4 |- H = ( LHyp ` K )
22		cdlemk1.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
23		cdlemk1.r	. . . 4 |- R = ( ( trL ` K ) ` W )
24		cdlemk1.s	. . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
25		cdlemk1.o	. . . 4 |- O = ( S ` D )
26		cdlemk1.u	. . . 4 |- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
27	16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26	cdlemkuv2 35538	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ C e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` C ) ` P ) = ( ( P .\/ ( R ` C ) ) ./\ ( ( O ` P ) .\/ ( R ` ( C o. `' D ) ) ) ) )
28	1, 2, 3, 4, 5, 6, 10, 14, 15, 27	syl333anc 1255	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( U ` C ) ` P ) = ( ( P .\/ ( R ` C ) ) ./\ ( ( O ` P ) .\/ ( R ` ( C o. `' D ) ) ) ) )
29	17, 18, 20, 21, 22, 23	trljat1 34837	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ C e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` C ) ) = ( P .\/ ( C ` P ) ) )
30	1, 3, 15, 29	syl3anc 1223	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( R ` C ) ) = ( P .\/ ( C ` P ) ) )
31	25	fveq1i 5858	. . . . 5 |- ( O ` P ) = ( ( S ` D ) ` P )
32	31	a1i 11	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( O ` P ) = ( ( S ` D ) ` P ) )
33	21, 22, 23	trlcocnv 35391	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ C e. T /\ D e. T ) -> ( R ` ( C o. `' D ) ) = ( R ` ( D o. `' C ) ) )
34	1, 3, 5, 33	syl3anc 1223	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( R ` ( C o. `' D ) ) = ( R ` ( D o. `' C ) ) )
35	32, 34	oveq12d 6293	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( O ` P ) .\/ ( R ` ( C o. `' D ) ) ) = ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' C ) ) ) )
36	30, 35	oveq12d 6293	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( R ` C ) ) ./\ ( ( O ` P ) .\/ ( R ` ( C o. `' D ) ) ) ) = ( ( P .\/ ( C ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' C ) ) ) ) )
37		cdlemk2a.q	. . . 4 |- Q = ( S ` C )
38	37	fveq1i 5858	. . 3 |- ( Q ` P ) = ( ( S ` C ) ` P )
39	6, 5	jca 532	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( N e. T /\ D e. T ) )
40		simp3r2 1100	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( R ` C ) =/= ( R ` F ) )
41	40, 7	jca 532	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
42	16, 17, 18, 20, 21, 22, 23, 19, 24	cdlemk12 35521	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( ( N e. T /\ D e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` C ) =/= ( R ` D ) ) ) -> ( ( S ` C ) ` P ) = ( ( P .\/ ( C ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' C ) ) ) ) )
43	1, 4, 3, 39, 15, 2, 14, 41, 8, 42	syl333anc 1255	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( S ` C ) ` P ) = ( ( P .\/ ( C ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' C ) ) ) ) )
44	38, 43	syl5req 2514	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( C ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' C ) ) ) ) = ( Q ` P ) )
45	28, 36, 44	3eqtrd 2505	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( U ` C ) ` P ) = ( Q ` P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2655   class class class wbr 4440   |-> cmpt 4498   _I cid 4783   `'ccnv 4991   |` cres 4994   o. ccom 4996   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830

This theorem is referenced by:  cdlemk20-2N  35563  cdlemk22  35564