Theorem cdlemk20 34515

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 1018	. . 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 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 ) ) ) ) -> ( R ` F ) = ( R ` N ) )
3		simp21r 1106	. . 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 1019	. . 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 1020	. . 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 1105	. . 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 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 ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
8		simp3r3 1098	. . . . 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 2693	. . . 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 1093	. . . 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 1095	. . . 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 1094	. . . 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 1168	. . 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 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 ) ) ) ) -> ( 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 34508	. . 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 1250	. 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 33807	. . . 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 1218	. . 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 5690	. . . . 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 34361	. . . . 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 1218	. . . 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 6107	. . 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 6107	. 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 5690	. . 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 1097	. . . . 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 34491	. . . 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 1250	. . 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 2486	. 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 2477	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 965   = wceq 1369   e. wcel 1756   =/= wne 2604   class class class wbr 4290   e. cmpt 4348   _I cid 4629   `'ccnv 4837   |` cres 4840   o. ccom 4842   ` cfv 5416   iota_crio 6049  (class class class)co 6089   Basecbs 14172   lecple 14243   joincjn 15112   meetcmee 15113   Atomscatm 32905   HLchlt 32992   LHypclh 33625   LTrncltrn 33742   trLctrl 33799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-riotaBAD 32601

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-undef 6790  df-map 7214  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-llines 33139  df-lplanes 33140  df-lvols 33141  df-lines 33142  df-psubsp 33144  df-pmap 33145  df-padd 33437  df-lhyp 33629  df-laut 33630  df-ldil 33745  df-ltrn 33746  df-trl 33800

This theorem is referenced by:  cdlemk20-2N  34533  cdlemk22  34534