Theorem cdlemk20 36162

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 1091	. . 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 1096	. . 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 1179	. . 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 1092	. . 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 1093	. . 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 1178	. . 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 1169	. . . 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 1171	. . . . 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 2849	. . . 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 554	. . 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 1166	. . . 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 1168	. . . 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 1167	. . . 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 1242	. . 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 1095	. . 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 36155	. . 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 1358	. 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 35453	. . . 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 1326	. . 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 6192	. . . . 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 36008	. . . . 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 1326	. . . 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 6668	. . 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 6668	. 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 6192	. . 3 |- ( Q ` P ) = ( ( S ` C ) ` P )
39	6, 5	jca 554	. . . 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 1170	. . . . 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 554	. . . 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 36138	. . . 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 1358	. . 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 2669	. 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 2660	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-undef 7399  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemk20-2N  36180  cdlemk22  36181