Theorem cdlemk53 36826

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 7, p. 120. G, I stand for g, h. X represents tau. (Contributed by NM, 26-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk5.b	|- B = ( Base ` K )
cdlemk5.l	|- .<_ = ( le ` K )
cdlemk5.j	|- .\/ = ( join ` K )
cdlemk5.m	|- ./\ = ( meet ` K )
cdlemk5.a	|- A = ( Atoms ` K )
cdlemk5.h	|- H = ( LHyp ` K )
cdlemk5.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r	|- R = ( ( trL ` K ) ` W )
cdlemk5.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )

Assertion

Ref	Expression
cdlemk53	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )

Distinct variable groups:   ./\ , g   .\/ , g   B, g   P, g   R, g   T, g   g, Z   g, b, G, z   ./\ , b, z   .<_ , b   z, g, .<_   .\/ , b, z   A, b, g, z   B, b, z   F, b, g, z   z, G   H, b, g, z   K, b, g, z   N, b, g, z   P, b, z   R, b, z   T, b, z   W, b, g, z   z, Y   G, b   I, b, g, z

Allowed substitution hints:   X( z, g, b)   Y( g, b)   Z( z, b)

Proof of Theorem cdlemk53

Step	Hyp	Ref	Expression
1		simp1l 1020	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp211 1134	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> F e. T )
3		simp212 1135	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> F =/= ( _I |` B ) )
4	2, 3	jca 532	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
5		simp22 1030	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> G e. T )
6		simp213 1136	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> N e. T )
7		simp23 1031	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( P e. A /\ -. P .<_ W ) )
8		simp1r 1021	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` F ) = ( R ` N ) )
9		cdlemk5.b	. . . . . . . 8 |- B = ( Base ` K )
10		cdlemk5.l	. . . . . . . 8 |- .<_ = ( le ` K )
11		cdlemk5.j	. . . . . . . 8 |- .\/ = ( join ` K )
12		cdlemk5.m	. . . . . . . 8 |- ./\ = ( meet ` K )
13		cdlemk5.a	. . . . . . . 8 |- A = ( Atoms ` K )
14		cdlemk5.h	. . . . . . . 8 |- H = ( LHyp ` K )
15		cdlemk5.t	. . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
16		cdlemk5.r	. . . . . . . 8 |- R = ( ( trL ` K ) ` W )
17		cdlemk5.z	. . . . . . . 8 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
18		cdlemk5.y	. . . . . . . 8 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
19		cdlemk5.x	. . . . . . . 8 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
20	9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	cdlemk35s-id 36807	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ G / g ]_ X e. T )
21	1, 4, 5, 6, 7, 8, 20	syl132anc 1246	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ G / g ]_ X e. T )
22	9, 14, 15	ltrn1o 35991	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T ) -> [_ G / g ]_ X : B -1-1-onto-> B )
23	1, 21, 22	syl2anc 661	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ G / g ]_ X : B -1-1-onto-> B )
24		f1of 5822	. . . . 5 |- ( [_ G / g ]_ X : B -1-1-onto-> B -> [_ G / g ]_ X : B --> B )
25		fcoi1 5765	. . . . 5 |- ( [_ G / g ]_ X : B --> B -> ( [_ G / g ]_ X o. ( _I |` B ) ) = [_ G / g ]_ X )
26	23, 24, 25	3syl 20	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X o. ( _I |` B ) ) = [_ G / g ]_ X )
27	26	adantr 465	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( [_ G / g ]_ X o. ( _I |` B ) ) = [_ G / g ]_ X )
28		simpl1l 1047	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
29	2, 6, 8	3jca 1176	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) )
30	29	adantr 465	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) )
31		simpl23 1076	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( P e. A /\ -. P .<_ W ) )
32		simpr 461	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> I = ( _I |` B ) )
33	9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	cdlemkid 36805	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ I = ( _I |` B ) ) ) -> [_ I / g ]_ X = ( _I |` B ) )
34	28, 30, 31, 32, 33	syl112anc 1232	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> [_ I / g ]_ X = ( _I |` B ) )
35	34	coeq2d 5175	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( [_ G / g ]_ X o. [_ I / g ]_ X ) = ( [_ G / g ]_ X o. ( _I |` B ) ) )
36	32	coeq2d 5175	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( G o. I ) = ( G o. ( _I |` B ) ) )
37	9, 14, 15	ltrn1o 35991	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> G : B -1-1-onto-> B )
38	1, 5, 37	syl2anc 661	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> G : B -1-1-onto-> B )
39		f1of 5822	. . . . . . 7 |- ( G : B -1-1-onto-> B -> G : B --> B )
40		fcoi1 5765	. . . . . . 7 |- ( G : B --> B -> ( G o. ( _I |` B ) ) = G )
41	38, 39, 40	3syl 20	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G o. ( _I |` B ) ) = G )
42	41	adantr 465	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( G o. ( _I |` B ) ) = G )
43	36, 42	eqtrd 2498	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> ( G o. I ) = G )
44	43	csbeq1d 3437	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> [_ ( G o. I ) / g ]_ X = [_ G / g ]_ X )
45	27, 35, 44	3eqtr4rd 2509	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I = ( _I |` B ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
46		simpl1 999	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I =/= ( _I |` B ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
47		simpl2 1000	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I =/= ( _I |` B ) ) -> ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) )
48		simpl3l 1051	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I =/= ( _I |` B ) ) -> I e. T )
49		simpr 461	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I =/= ( _I |` B ) ) -> I =/= ( _I |` B ) )
50		simpl3r 1052	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I =/= ( _I |` B ) ) -> ( R ` G ) =/= ( R ` I ) )
51	9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	cdlemk53b 36825	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
52	46, 47, 48, 49, 50, 51	syl113anc 1240	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) /\ I =/= ( _I |` B ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
53	45, 52	pm2.61dane 2775	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   [_csb 3430   class class class wbr 4456   _I cid 4799   `'ccnv 5007   |` cres 5010   o. ccom 5012   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   Atomscatm 35131   HLchlt 35218   LHypclh 35851   LTrncltrn 35968   trLctrl 36026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027

This theorem is referenced by:  cdlemk54  36827  cdlemk55  36830