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Theorem cdlemc1 30827
Description: Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 30500? (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 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 hllat 30000 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
4 simp3l 985 . . . 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 29926 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
84, 7syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  B )
9 simp2 958 . . . . 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 14467 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  e.  B )
123, 8, 9, 11syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  X )  e.  B
)
13 simp1r 982 . . . . 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 30634 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
1613, 15syl 16 . . . 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 14468 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  X
)  ./\  W )  e.  B )
193, 12, 16, 18syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  W )  e.  B )
205, 10latjcom 14476 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( ( P  .\/  X )  ./\  W )  e.  B )  ->  ( P  .\/  ( ( P 
.\/  X )  ./\  W ) )  =  ( ( ( P  .\/  X )  ./\  W )  .\/  P ) )
213, 8, 19, 20syl3anc 1184 . 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 14477 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  P  .<_  ( P  .\/  X ) )
243, 8, 9, 23syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  .<_  ( P  .\/  X ) )
255, 22, 10, 17, 6atmod2i1 30497 . . 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 1197 . 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 2435 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2822, 10, 27, 6, 14lhpjat1 30656 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
29283adant2 976 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  ( 1. `  K ) )
3029oveq2d 6088 . . 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 29998 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
321, 31syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OL )
335, 17, 27olm11 29864 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  X )  e.  B )  -> 
( ( P  .\/  X )  ./\  ( 1. `  K ) )  =  ( P  .\/  X
) )
3432, 12, 33syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  ( 1. `  K ) )  =  ( P 
.\/  X ) )
3530, 34eqtrd 2467 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  ( W  .\/  P ) )  =  ( P 
.\/  X ) )
3621, 26, 353eqtrd 2471 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   1.cp1 14455   Latclat 14462   OLcol 29811   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdlemc2  30828  cdlemd1  30834
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 4692
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-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624
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