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Theorem cdlemk5a 36977
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 1018 . . 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 1019 . . 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 1027 . . 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 1028 . . 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 996 . . 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 36975 . . 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 1237 . 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 1029 . . 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 1121 . . 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 1122 . . 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 36976 . . 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 1242 . 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 4459 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   _I cid 4779   `'ccnv 4987   |` cres 4990   o. ccom 4992   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299
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 35100
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 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300
This theorem is referenced by:  cdlemk5  36978  cdlemk5auN  37002  cdlemk5u  37003
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