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Theorem cdlemk6u 36985
Description: Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 36007. (Contributed by NM, 4-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 )
Assertion
Ref Expression
cdlemk6u |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` 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
Allowed substitution hints:   A( f)   B( f, i)   S( f, i)   G( f, i)   H( f)   K( f)   .<_ ( f)   O( f, i)   X( f, i)
Proof of Theorem cdlemk6u
StepHypRef Expression
1 cdlemk1.b . . 3 |- B = ( Base ` K )
2 cdlemk1.l . . 3 |- .<_ = ( le ` K )
3 cdlemk1.j . . 3 |- .\/ = ( join ` K )
4 cdlemk1.m . . 3 |- ./\ = ( meet ` K )
5 cdlemk1.a . . 3 |- A = ( Atoms ` K )
6 cdlemk1.h . . 3 |- H = ( LHyp ` K )
7 cdlemk1.t . . 3 |- T = ( ( LTrn ` K ) ` W )
8 cdlemk1.r . . 3 |- R = ( ( trL ` K ) ` W )
9 cdlemk1.s . . 3 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk1.o . . 3 |- O = ( S ` D )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cdlemk5u 36984 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) )
12 simp11l 1105 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> K e. HL )
13 simp22l 1113 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> P e. A )
14 simp11 1024 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp212 1133 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> G e. T )
162, 5, 6, 7ltrnat 36261 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
1714, 15, 13, 16syl3anc 1226 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( G ` P ) e. A )
18 simp213 1134 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> X e. T )
192, 5, 6, 7ltrnat 36261 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A )
2014, 18, 13, 19syl3anc 1226 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( X ` P ) e. A )
21 simp1 994 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
22 simp211 1132 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> N e. T )
23 simp22 1028 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
24 simp23 1029 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( R ` F ) = ( R ` N ) )
25 simp3l1 1099 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> F =/= ( _I |` B ) )
26 simp3l2 1100 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> D =/= ( _I |` B ) )
27 simp3r1 1102 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
281, 2, 3, 4, 5, 6, 7, 8, 9, 10cdlemkoatnle 36974 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )
2928simpld 457 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) e. A )
3021, 22, 23, 24, 25, 26, 27, 29syl133anc 1249 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( O ` P ) e. A )
31 simp13 1026 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> D e. T )
32 simp3r2 1103 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
335, 6, 7, 8trlcocnvat 36847 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ D e. T ) /\ ( R ` G ) =/= ( R ` D ) ) -> ( R ` ( G o. `' D ) ) e. A )
3414, 15, 31, 32, 33syl121anc 1231 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( G o. `' D ) ) e. A )
35 simp3r3 1104 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( R ` X ) =/= ( R ` D ) )
365, 6, 7, 8trlcocnvat 36847 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ D e. T ) /\ ( R ` X ) =/= ( R ` D ) ) -> ( R ` ( X o. `' D ) ) e. A )
3714, 18, 31, 35, 36syl121anc 1231 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( X o. `' D ) ) e. A )
382, 3, 4, 5dalaw 36007 . . 3 |- ( ( K e. HL /\ ( P e. A /\ ( G ` P ) e. A /\ ( X ` P ) e. A ) /\ ( ( O ` P ) e. A /\ ( R ` ( G o. `' D ) ) e. A /\ ( R ` ( X o. `' D ) ) e. A ) ) -> ( ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) ) )
3912, 13, 17, 20, 30, 34, 37, 38syl133anc 1249 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) ) )
4011, 39mpd 15 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X 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 ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   |-> cmpt 4497   _I cid 4779   `'ccnv 4987   |` cres 4990   o. ccom 4992   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   trLctrl 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281
This theorem is referenced by:  cdlemk7u  36993
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