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Theorem cdlemc4 30722
Description: Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l  |-  .<_  =  ( le `  K )
cdlemc3.j  |-  .\/  =  ( join `  K )
cdlemc3.m  |-  ./\  =  ( meet `  K )
cdlemc3.a  |-  A  =  ( Atoms `  K )
cdlemc3.h  |-  H  =  ( LHyp `  K
)
cdlemc3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemc3.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemc4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P )
) )  ->  ( Q  .\/  ( R `  F ) )  =/=  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )

Proof of Theorem cdlemc4
StepHypRef Expression
1 simpll 731 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  HL )
2 hllat 29892 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  Lat )
4 simpl 444 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5 simpr1 963 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  F  e.  T )
6 simpr2l 1016 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  P  e.  A )
7 eqid 2430 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
8 cdlemc3.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
97, 8atbase 29818 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
106, 9syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  P  e.  ( Base `  K
) )
11 cdlemc3.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
12 cdlemc3.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
137, 11, 12ltrncl 30653 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
144, 5, 10, 13syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  P )  e.  ( Base `  K
) )
15 simpr3l 1018 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  e.  A )
16 cdlemc3.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
177, 16, 8hlatjcl 29895 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
181, 6, 15, 17syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
197, 11lhpbase 30526 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2019ad2antlr 708 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  W  e.  ( Base `  K
) )
21 cdlemc3.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
227, 21latmcl 14463 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
233, 18, 20, 22syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  W )  e.  ( Base `  K
) )
24 cdlemc3.l . . . . . . 7  |-  .<_  =  ( le `  K )
257, 24, 16latlej1 14472 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( F `  P )  e.  ( Base `  K
)  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )  -> 
( F `  P
)  .<_  ( ( F `
 P )  .\/  ( ( P  .\/  Q )  ./\  W )
) )
263, 14, 23, 25syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  P )  .<_  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )
27 breq2 4203 . . . . 5  |-  ( ( Q  .\/  ( R `
 F ) )  =  ( ( F `
 P )  .\/  ( ( P  .\/  Q )  ./\  W )
)  ->  ( ( F `  P )  .<_  ( Q  .\/  ( R `  F )
)  <->  ( F `  P )  .<_  ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) ) )
2826, 27syl5ibrcom 214 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( Q  .\/  ( R `  F )
)  =  ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ->  ( F `  P )  .<_  ( Q  .\/  ( R `  F )
) ) )
29 cdlemc3.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
3024, 16, 21, 8, 11, 12, 29cdlemc3 30721 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .<_  ( Q  .\/  ( R `  F ) )  ->  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )
3128, 30syld 42 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( Q  .\/  ( R `  F )
)  =  ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ->  Q  .<_  ( P  .\/  ( F `  P )
) ) )
3231necon3bd 2630 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( -.  Q  .<_  ( P 
.\/  ( F `  P ) )  -> 
( Q  .\/  ( R `  F )
)  =/=  ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) ) )
33323impia 1150 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P )
) )  ->  ( Q  .\/  ( R `  F ) )  =/=  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   joincjn 14384   meetcmee 14385   Latclat 14457   Atomscatm 29792   HLchlt 29879   LHypclh 30512   LTrncltrn 30629   trLctrl 30686
This theorem is referenced by:  cdlemc5  30723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-map 7006  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516  df-laut 30517  df-ldil 30632  df-ltrn 30633  df-trl 30687
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