Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk5a Structured version   Unicode version
Theorem cdlemk5a 34782
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
Hypotheses
Ref Expression
cdlemk.b |- B = ( Base ` K )
cdlemk.l |- .<_ = ( le ` K )
cdlemk.j |- .\/ = ( join ` K )
cdlemk.a |- A = ( Atoms ` K )
cdlemk.h |- H = ( LHyp ` K )
cdlemk.t |- T = ( ( LTrn ` K ) ` W )
cdlemk.r |- R = ( ( trL ` K ) ` W )
cdlemk.m |- ./\ = ( meet ` K )
Assertion
Ref Expression
cdlemk5a |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
Proof of Theorem cdlemk5a
StepHypRef Expression
1 simp1l 1012 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL )
2 simp1r 1013 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H )
3 simp21 1021 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
4 simp22 1022 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
5 simp3 990 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) )
6 cdlemk.b . . . 4 |- B = ( Base ` K )
7 cdlemk.l . . . 4 |- .<_ = ( le ` K )
8 cdlemk.j . . . 4 |- .\/ = ( join ` K )
9 cdlemk.a . . . 4 |- A = ( Atoms ` K )
10 cdlemk.h . . . 4 |- H = ( LHyp ` K )
11 cdlemk.t . . . 4 |- T = ( ( LTrn ` K ) ` W )
12 cdlemk.r . . . 4 |- R = ( ( trL ` K ) ` W )
13 cdlemk.m . . . 4 |- ./\ = ( meet ` K )
146, 7, 8, 9, 10, 11, 12, 13cdlemk3 34780 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( F ` P ) )
151, 2, 3, 4, 5, 14syl221anc 1230 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( F ` P ) )
16 simp23 1023 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X e. T )
17 simp33l 1115 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> P e. A )
18 simp33r 1116 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> -. P .<_ W )
196, 7, 8, 9, 10, 11, 12, 13cdlemk4 34781 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
201, 2, 3, 16, 17, 18, 19syl222anc 1235 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F ` P ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
2115, 20eqbrtrd 4407 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2642   class class class wbr 4387   _I cid 4726   `'ccnv 4934   |` cres 4937   o. ccom 4939   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   joincjn 15213   meetcmee 15214   Atomscatm 33211   HLchlt 33298   LHypclh 33931   LTrncltrn 34048   trLctrl 34105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-riotaBAD 32907
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-iin 4269  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-undef 6889  df-map 7313  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-p1 15309  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-llines 33445  df-lplanes 33446  df-lvols 33447  df-lines 33448  df-psubsp 33450  df-pmap 33451  df-padd 33743  df-lhyp 33935  df-laut 33936  df-ldil 34051  df-ltrn 34052  df-trl 34106
This theorem is referenced by:  cdlemk5  34783  cdlemk5auN  34807  cdlemk5u  34808
  Copyright terms: Public domain W3C validator