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Theorem cdlemg12a 31377
Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemg12a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  (
( P  .\/  U
)  ./\  ( ( G `  P )  .\/  U ) )  .<_  ( ( F `  ( G `  P ) )  .\/  U ) )

Proof of Theorem cdlemg12a
StepHypRef Expression
1 simp1l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  K  e.  HL )
2 simp21l 1074 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  P  e.  A )
3 simp1 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp31 993 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  G  e.  T )
5 cdlemg12.l . . . . 5  |-  .<_  =  ( le `  K )
6 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
7 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnat 30874 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
103, 4, 2, 9syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  ( G `  P )  e.  A )
11 simp1r 982 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  W  e.  H )
12 simp21 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
13 simp22l 1076 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  Q  e.  A )
14 simp32 994 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  P  =/=  Q )
15 cdlemg12.j . . . . 5  |-  .\/  =  ( join `  K )
16 cdlemg12.m . . . . 5  |-  ./\  =  ( meet `  K )
17 cdlemg12.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
185, 15, 16, 6, 7, 17cdleme0a 30945 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
191, 11, 12, 13, 14, 18syl212anc 1194 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  U  e.  A )
20 simp33 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  ( P  .\/  U )  =/=  ( ( G `  P )  .\/  U
) )
215, 15, 16, 62llnma3r 30522 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( G `  P
)  e.  A  /\  U  e.  A )  /\  ( P  .\/  U
)  =/=  ( ( G `  P ) 
.\/  U ) )  ->  ( ( P 
.\/  U )  ./\  ( ( G `  P )  .\/  U
) )  =  U )
221, 2, 10, 19, 20, 21syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  (
( P  .\/  U
)  ./\  ( ( G `  P )  .\/  U ) )  =  U )
23 simp23 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  F  e.  T )
245, 6, 7, 8ltrncoat 30878 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
253, 23, 4, 2, 24syl121anc 1189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  ( F `  ( G `  P ) )  e.  A )
265, 15, 6hlatlej2 30110 . . 3  |-  ( ( K  e.  HL  /\  ( F `  ( G `
 P ) )  e.  A  /\  U  e.  A )  ->  U  .<_  ( ( F `  ( G `  P ) )  .\/  U ) )
271, 25, 19, 26syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  U  .<_  ( ( F `  ( G `  P ) )  .\/  U ) )
2822, 27eqbrtrd 4224 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( P  .\/  U )  =/=  (
( G `  P
)  .\/  U )
) )  ->  (
( P  .\/  U
)  ./\  ( ( G `  P )  .\/  U ) )  .<_  ( ( F `  ( G `  P ) )  .\/  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
This theorem is referenced by:  cdlemg12b  31378
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839
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