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Theorem cdlema2N 29148
Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlema2.b  |-  B  =  ( Base `  K
)
cdlema2.l  |-  .<_  =  ( le `  K )
cdlema2.j  |-  .\/  =  ( join `  K )
cdlema2.m  |-  ./\  =  ( meet `  K )
cdlema2.z  |-  .0.  =  ( 0. `  K )
cdlema2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlema2N  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  ./\  X
)  =  .0.  )

Proof of Theorem cdlema2N
StepHypRef Expression
1 simp3ll 1031 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  =/=  P )
2 simp3rl 1033 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  P  .<_  X )
3 simp3rr 1034 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  -.  Q  .<_  X )
4 simp3lr 1032 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  .<_  ( P  .\/  Q ) )
52, 3, 43jca 1137 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )
6 cdlema2.b . . . . . 6  |-  B  =  ( Base `  K
)
7 cdlema2.l . . . . . 6  |-  .<_  =  ( le `  K )
8 cdlema2.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdlema2.a . . . . . 6  |-  A  =  ( Atoms `  K )
106, 7, 8, 9exatleN 28760 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
115, 10syld3an3 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
1211necon3bbid 2455 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( -.  R  .<_  X  <-> 
R  =/=  P ) )
131, 12mpbird 225 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  -.  R  .<_  X )
14 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  K  e.  HL )
15 hlatl 28717 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
1614, 15syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  K  e.  AtLat )
17 simp23 995 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  e.  A )
18 simp1r 985 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  X  e.  B )
19 cdlema2.m . . . 4  |-  ./\  =  ( meet `  K )
20 cdlema2.z . . . 4  |-  .0.  =  ( 0. `  K )
216, 7, 19, 20, 9atnle 28674 . . 3  |-  ( ( K  e.  AtLat  /\  R  e.  A  /\  X  e.  B )  ->  ( -.  R  .<_  X  <->  ( R  ./\ 
X )  =  .0.  ) )
2216, 17, 18, 21syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( -.  R  .<_  X  <-> 
( R  ./\  X
)  =  .0.  )
)
2313, 22mpbid 203 1  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  ./\  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   0.cp0 14105   Atomscatm 28620   AtLatcal 28621   HLchlt 28707
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-lat 14114  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708
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