Theorem cdlemk11u-2N 31371

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

Step	Hyp	Ref	Expression
1		simp11 987	. . 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 988	. . 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 )
3	1, 2	jca 519	. 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 1095	. 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 1096	. 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 1097	. . 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 1076	. . 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 1078	. . 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 )
9	6, 7, 8	3jca 1134	. 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 995	. 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 989	. 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 1082	. . 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 1083	. . 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 1077	. . 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 ) )
15	12, 13, 14	3jca 1134	. 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 1079	. 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 993	. 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 ) ) ) )
30	18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29	cdlemk11u 31353	. 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 ) ) ) )
31	3, 4, 5, 9, 10, 11, 15, 16, 17, 30	syl333anc 1216	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 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   class class class wbr 4172   e. cmpt 4226   _I cid 4453   `'ccnv 4836   |` cres 4839   o. ccom 4841   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641