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Theorem cdlemg12a 30832
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 979 . . 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 1072 . . 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 955 . . . 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 991 . . . 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 30329 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
103, 4, 2, 9syl3anc 1182 . . 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 980 . . . 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 988 . . . 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 1074 . . . 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 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 )
) )  ->  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 30400 . . . 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 1192 . . 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 993 . . 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 29977 . . 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 1195 . 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 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 )
) )  ->  F  e.  T )
245, 6, 7, 8ltrncoat 30333 . . . 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 1187 . . 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 29565 . . 3  |-  ( ( K  e.  HL  /\  ( F `  ( G `
 P ) )  e.  A  /\  U  e.  A )  ->  U  .<_  ( ( F `  ( G `  P ) )  .\/  U ) )
271, 25, 19, 26syl3anc 1182 . 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 4043 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  cdlemg12b  30833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294
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