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Theorem cdlemk11u-2N 34457
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma2 ( Z) case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk2.b |- B = ( Base ` K )
cdlemk2.l |- .<_ = ( le ` K )
cdlemk2.j |- .\/ = ( join ` K )
cdlemk2.m |- ./\ = ( meet ` K )
cdlemk2.a |- A = ( Atoms ` K )
cdlemk2.h |- H = ( LHyp ` K )
cdlemk2.t |- T = ( ( LTrn ` K ) ` W )
cdlemk2.r |- R = ( ( trL ` K ) ` W )
cdlemk2.s |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk2.q |- Q = ( S ` C )
cdlemk2.v |- V = ( d e. T |-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
cdlemk2.z |- Z = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' C ) ) .\/ ( R ` ( X o. `' C ) ) ) )
Assertion
Ref Expression
cdlemk11u-2N |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( V ` G ) ` P ) .<_ ( ( ( V ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   C, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i   ./\ , d   .\/ , d   C, d, k   G, d, k   Q, d   P, d   R, d   T, d   W, d   ./\ , k   .<_ , k   .\/ , k   A, k   C, k   k, F   k, H   k, K   k, N   Q, k   P, k   R, k   T, k   k, W   F, d   X, d, k
Allowed substitution hints:   A( f, d)   B( f, i, k, d)   Q( f, i)   S( f, i, k, d)   G( f, i)   H( f, d)   K( f, d)   .<_ ( f, d)   N( d)   V( f, i, k, d)   X( f, i)   Z( f, i, k, d)
Proof of Theorem cdlemk11u-2N
StepHypRef Expression
1 simp11 1039 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL )
2 simp12 1040 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H )
31, 2jca 539 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
4 simp211 1147 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
5 simp212 1148 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C e. T )
6 simp213 1149 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T )
7 simp22l 1128 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
8 simp23l 1130 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X e. T )
96, 7, 83jca 1189 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( N e. T /\ G e. T /\ X e. T ) )
10 simp33 1047 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp13 1041 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
12 simp32l 1134 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
13 simp32r 1135 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C =/= ( _I |` B ) )
14 simp22r 1129 . . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G =/= ( _I |` B ) )
1512, 13, 143jca 1189 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
16 simp23r 1131 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X =/= ( _I |` B ) )
17 simp31 1045 . 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) )
18 cdlemk2.b . . 3 |- B = ( Base ` K )
19 cdlemk2.l . . 3 |- .<_ = ( le ` K )
20 cdlemk2.j . . 3 |- .\/ = ( join ` K )
21 cdlemk2.m . . 3 |- ./\ = ( meet ` K )
22 cdlemk2.a . . 3 |- A = ( Atoms ` K )
23 cdlemk2.h . . 3 |- H = ( LHyp ` K )
24 cdlemk2.t . . 3 |- T = ( ( LTrn ` K ) ` W )
25 cdlemk2.r . . 3 |- R = ( ( trL ` K ) ` W )
26 cdlemk2.s . . 3 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
27 cdlemk2.q . . 3 |- Q = ( S ` C )
28 cdlemk2.v . . 3 |- V = ( d e. T |-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
29 cdlemk2.z . . 3 |- Z = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' C ) ) .\/ ( R ` ( X o. `' C ) ) ) )
3018, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29cdlemk11u 34439 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ X =/= ( _I |` B ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) ) ) -> ( ( V ` G ) ` P ) .<_ ( ( ( V ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
313, 4, 5, 9, 10, 11, 15, 16, 17, 30syl333anc 1303 1 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( V ` G ) ` P ) .<_ ( ( ( V ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   /\ w3a 986   = wceq 1447   e. wcel 1890   =/= wne 2621   class class class wbr 4373   |-> cmpt 4432   _I cid 4721   `'ccnv 4810   |` cres 4813   o. ccom 4815   ` cfv 5560   iota_crio 6236  (class class class)co 6275   Basecbs 15131   lecple 15207   joincjn 16199   meetcmee 16200   Atomscatm 32830   HLchlt 32917   LHypclh 33550   LTrncltrn 33667   trLctrl 33725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570  ax-riotaBAD 32526
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-iin 4250  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6780  df-2nd 6781  df-undef 7006  df-map 7460  df-preset 16183  df-poset 16201  df-plt 16214  df-lub 16230  df-glb 16231  df-join 16232  df-meet 16233  df-p0 16295  df-p1 16296  df-lat 16302  df-clat 16364  df-oposet 32743  df-ol 32745  df-oml 32746  df-covers 32833  df-ats 32834  df-atl 32865  df-cvlat 32889  df-hlat 32918  df-llines 33064  df-lplanes 33065  df-lvols 33066  df-lines 33067  df-psubsp 33069  df-pmap 33070  df-padd 33362  df-lhyp 33554  df-laut 33555  df-ldil 33670  df-ltrn 33671  df-trl 33726
This theorem is referenced by: (None)
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