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Theorem cdlemk5a 34447
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 1038 . . 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 1039 . . 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 1047 . . 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 1048 . . 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 1016 . . 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 34445 . . 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 1287 . 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 1049 . . 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 1141 . . 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 1142 . . 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 34446 . . 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 1292 . 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 4437 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 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   class class class wbr 4416   _I cid 4763   `'ccnv 4852   |` cres 4855   o. ccom 4857   ` cfv 5601  (class class class)co 6315   Basecbs 15170   lecple 15246   joincjn 16238   meetcmee 16239   Atomscatm 32874   HLchlt 32961   LHypclh 33594   LTrncltrn 33711   trLctrl 33769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-riotaBAD 32570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-undef 7046  df-map 7500  df-preset 16222  df-poset 16240  df-plt 16253  df-lub 16269  df-glb 16270  df-join 16271  df-meet 16272  df-p0 16334  df-p1 16335  df-lat 16341  df-clat 16403  df-oposet 32787  df-ol 32789  df-oml 32790  df-covers 32877  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962  df-llines 33108  df-lplanes 33109  df-lvols 33110  df-lines 33111  df-psubsp 33113  df-pmap 33114  df-padd 33406  df-lhyp 33598  df-laut 33599  df-ldil 33714  df-ltrn 33715  df-trl 33770
This theorem is referenced by:  cdlemk5  34448  cdlemk5auN  34472  cdlemk5u  34473
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