Theorem cdlemk21N 34357

Description: Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=0 and j=1. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)

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 ) ) ) ) ) )

Assertion

Ref	Expression
cdlemk21N	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( U ` G ) ` 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, G, 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   i, G, f

Allowed substitution hints:   A( e, f)   B( 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 cdlemk21N

Step	Hyp	Ref	Expression
1		simp11 1018	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp21r 1106	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> G e. T )
3		simp22 1022	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
4		cdlemk1.l	. . . . 5 |- .<_ = ( le ` K )
5		cdlemk1.j	. . . . 5 |- .\/ = ( join ` K )
6		cdlemk1.a	. . . . 5 |- A = ( Atoms ` K )
7		cdlemk1.h	. . . . 5 |- H = ( LHyp ` K )
8		cdlemk1.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
9		cdlemk1.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
10	4, 5, 6, 7, 8, 9	trljat1 33650	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
11	1, 2, 3, 10	syl3anc 1218	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
12		cdlemk1.o	. . . . . 6 |- O = ( S ` D )
13	12	fveq1i 5687	. . . . 5 |- ( O ` P ) = ( ( S ` D ) ` P )
14	13	a1i 11	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( O ` P ) = ( ( S ` D ) ` P ) )
15		simp13 1020	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> D e. T )
16	7, 8, 9	trlcocnv 34204	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ D e. T ) -> ( R ` ( G o. `' D ) ) = ( R ` ( D o. `' G ) ) )
17	1, 2, 15, 16	syl3anc 1218	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` ( G o. `' D ) ) = ( R ` ( D o. `' G ) ) )
18	14, 17	oveq12d 6104	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) = ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) )
19	11, 18	oveq12d 6104	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) )
20		simp23 1023	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` F ) = ( R ` N ) )
21		simp12 1019	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> F e. T )
22		simp21l 1105	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> N e. T )
23		simp3r1 1096	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
24		simp3r2 1097	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
25	24	necomd 2690	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` D ) =/= ( R ` G ) )
26	23, 25	jca 532	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) )
27		simp3l1 1093	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> F =/= ( _I |` B ) )
28		simp3l3 1095	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> G =/= ( _I |` B ) )
29		simp3l2 1094	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> D =/= ( _I |` B ) )
30	27, 28, 29	3jca 1168	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) )
31		cdlemk1.b	. . . 4 |- B = ( Base ` K )
32		cdlemk1.m	. . . 4 |- ./\ = ( meet ` K )
33		cdlemk1.s	. . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
34		cdlemk1.u	. . . 4 |- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
35	31, 4, 5, 32, 6, 7, 8, 9, 33, 12, 34	cdlemkuv2 34351	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
36	1, 20, 2, 21, 15, 22, 26, 30, 3, 35	syl333anc 1250	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
37	22, 15	jca 532	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( N e. T /\ D e. T ) )
38		simp3r3 1098	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` F ) )
39	38, 23	jca 532	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( R ` G ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
40	31, 4, 5, 6, 7, 8, 9, 32, 33	cdlemk12 34334	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ D e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) )
41	1, 21, 2, 37, 3, 20, 30, 39, 24, 40	syl333anc 1250	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) )
42	19, 36, 41	3eqtr4rd 2481	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( U ` G ) ` P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   class class class wbr 4287   e. cmpt 4345   _I cid 4626   `'ccnv 4834   |` cres 4837   o. ccom 4839   ` cfv 5413   iota_crio 6046  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   Atomscatm 32748   HLchlt 32835   LHypclh 33468   LTrncltrn 33585   trLctrl 33642

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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32444

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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-undef 6784  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643

This theorem is referenced by:  cdlemk21-2N  34375