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Theorem cdlemc1 29647
Description: Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 29320? (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemc1.b  |-  B  =  ( Base `  K
)
cdlemc1.l  |-  .<_  =  ( le `  K )
cdlemc1.j  |-  .\/  =  ( join `  K )
cdlemc1.m  |-  ./\  =  ( meet `  K )
cdlemc1.a  |-  A  =  ( Atoms `  K )
cdlemc1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemc1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  X )  ./\  W )
)  =  ( P 
.\/  X ) )

Proof of Theorem cdlemc1
StepHypRef Expression
1 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 hllat 28820 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
4 simp3l 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
5 cdlemc1.b . . . . 5  |-  B  =  ( Base `  K
)
6 cdlemc1.a . . . . 5  |-  A  =  ( Atoms `  K )
75, 6atbase 28746 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
84, 7syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  B )
9 simp2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  X  e.  B )
10 cdlemc1.j . . . . . 6  |-  .\/  =  ( join `  K )
115, 10latjcl 14150 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  e.  B )
123, 8, 9, 11syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  X )  e.  B
)
13 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
14 cdlemc1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
155, 14lhpbase 29454 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
1613, 15syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  B )
17 cdlemc1.m . . . . 5  |-  ./\  =  ( meet `  K )
185, 17latmcl 14151 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  X
)  ./\  W )  e.  B )
193, 12, 16, 18syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  W )  e.  B )
205, 10latjcom 14159 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( ( P  .\/  X )  ./\  W )  e.  B )  ->  ( P  .\/  ( ( P 
.\/  X )  ./\  W ) )  =  ( ( ( P  .\/  X )  ./\  W )  .\/  P ) )
213, 8, 19, 20syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  X )  ./\  W )
)  =  ( ( ( P  .\/  X
)  ./\  W )  .\/  P ) )
22 cdlemc1.l . . . . 5  |-  .<_  =  ( le `  K )
235, 22, 10latlej1 14160 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  P  .<_  ( P  .\/  X ) )
243, 8, 9, 23syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  .<_  ( P  .\/  X ) )
255, 22, 10, 17, 6atmod2i1 29317 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  X
)  e.  B  /\  W  e.  B )  /\  P  .<_  ( P 
.\/  X ) )  ->  ( ( ( P  .\/  X ) 
./\  W )  .\/  P )  =  ( ( P  .\/  X ) 
./\  ( W  .\/  P ) ) )
261, 4, 12, 16, 24, 25syl131anc 1200 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( P  .\/  X
)  ./\  W )  .\/  P )  =  ( ( P  .\/  X
)  ./\  ( W  .\/  P ) ) )
27 eqid 2284 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2822, 10, 27, 6, 14lhpjat1 29476 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
29283adant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  ( 1. `  K ) )
3029oveq2d 5835 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  ( W  .\/  P ) )  =  ( ( P  .\/  X ) 
./\  ( 1. `  K ) ) )
31 hlol 28818 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
321, 31syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OL )
335, 17, 27olm11 28684 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  X )  e.  B )  -> 
( ( P  .\/  X )  ./\  ( 1. `  K ) )  =  ( P  .\/  X
) )
3432, 12, 33syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  ( 1. `  K ) )  =  ( P 
.\/  X ) )
3530, 34eqtrd 2316 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  ( W  .\/  P ) )  =  ( P 
.\/  X ) )
3621, 26, 353eqtrd 2320 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  X )  ./\  W )
)  =  ( P 
.\/  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   1.cp1 14138   Latclat 14145   OLcol 28631   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem is referenced by:  cdlemc2  29648  cdlemd1  29654
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444
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