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Theorem lhpat 30303
Description: Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
Hypotheses
Ref Expression
lhpat.l  |-  .<_  =  ( le `  K )
lhpat.j  |-  .\/  =  ( join `  K )
lhpat.m  |-  ./\  =  ( meet `  K )
lhpat.a  |-  A  =  ( Atoms `  K )
lhpat.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpat  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P 
.\/  Q )  ./\  W )  e.  A )

Proof of Theorem lhpat
StepHypRef Expression
1 simp1l 980 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  K  e.  HL )
2 simp2l 982 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  P  e.  A
)
3 simp3l 984 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  Q  e.  A
)
4 simp1r 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  W  e.  H
)
5 eqid 2366 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 lhpat.h . . . 4  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 30258 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
84, 7syl 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  W  e.  (
Base `  K )
)
9 simp3r 985 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  P  =/=  Q
)
10 eqid 2366 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
11 eqid 2366 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
1210, 11, 6lhp1cvr 30259 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
) ( 1. `  K ) )
13123ad2ant1 977 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  W (  <o  `  K ) ( 1.
`  K ) )
14 simp2r 983 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  -.  P  .<_  W )
15 lhpat.l . . 3  |-  .<_  =  ( le `  K )
16 lhpat.j . . 3  |-  .\/  =  ( join `  K )
17 lhpat.m . . 3  |-  ./\  =  ( meet `  K )
18 lhpat.a . . 3  |-  A  =  ( Atoms `  K )
195, 15, 16, 17, 10, 11, 181cvrat 29736 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  W  e.  ( Base `  K ) )  /\  ( P  =/= 
Q  /\  W (  <o  `  K ) ( 1. `  K )  /\  -.  P  .<_  W ) )  ->  (
( P  .\/  Q
)  ./\  W )  e.  A )
201, 2, 3, 8, 9, 13, 14, 19syl133anc 1206 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P 
.\/  Q )  ./\  W )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   1.cp1 14354    <o ccvr 29523   Atomscatm 29524   HLchlt 29611   LHypclh 30244
This theorem is referenced by:  lhpat2  30305  4atexlemex6  30334  trlat  30429  cdlemc5  30455  cdleme3e  30492  cdleme7b  30504  cdleme11k  30528  cdleme16e  30542  cdleme16f  30543  cdlemeda  30558  cdleme22cN  30602  cdleme22d  30603  cdleme23b  30610  cdlemf2  30822  cdlemg12g  30909  cdlemg17dALTN  30924  cdlemg19a  30943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-lhyp 30248
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