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Theorem 1cvrat 29590
Description: Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
1cvrat.b  |-  B  =  ( Base `  K
)
1cvrat.l  |-  .<_  =  ( le `  K )
1cvrat.j  |-  .\/  =  ( join `  K )
1cvrat.m  |-  ./\  =  ( meet `  K )
1cvrat.u  |-  .1.  =  ( 1. `  K )
1cvrat.c  |-  C  =  (  <o  `  K )
1cvrat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
1cvrat  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )

Proof of Theorem 1cvrat
StepHypRef Expression
1 hllat 29478 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  K  e.  Lat )
3 simp21 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  e.  A )
4 1cvrat.b . . . . . . 7  |-  B  =  ( Base `  K
)
5 1cvrat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
64, 5atbase 29404 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  e.  B )
8 simp22 991 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  e.  A )
94, 5atbase 29404 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  e.  B )
11 1cvrat.j . . . . . 6  |-  .\/  =  ( join `  K )
124, 11latjcom 14415 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
132, 7, 10, 12syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( P  .\/  Q
)  =  ( Q 
.\/  P ) )
1413oveq1d 6035 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  =  ( ( Q 
.\/  P )  ./\  X ) )
154, 11latjcl 14406 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  P  e.  B )  ->  ( Q  .\/  P
)  e.  B )
162, 10, 7, 15syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( Q  .\/  P
)  e.  B )
17 simp23 992 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  X  e.  B )
18 1cvrat.m . . . . 5  |-  ./\  =  ( meet `  K )
194, 18latmcom 14431 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  P )  e.  B  /\  X  e.  B )  ->  (
( Q  .\/  P
)  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
202, 16, 17, 19syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( Q  .\/  P )  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
2114, 20eqtrd 2419 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
22 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  K  e.  HL )
2317, 8, 33jca 1134 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )
24 simp31 993 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  =/=  Q )
2524necomd 2633 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  =/=  P )
26 simp33 995 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  -.  P  .<_  X )
27 hlop 29477 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
28273ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  K  e.  OP )
29 1cvrat.l . . . . . 6  |-  .<_  =  ( le `  K )
30 1cvrat.u . . . . . 6  |-  .1.  =  ( 1. `  K )
314, 29, 30ople1 29306 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  B )  ->  Q  .<_  .1.  )
3228, 10, 31syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  .<_  .1.  )
33 simp32 994 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  X C  .1.  )
34 1cvrat.c . . . . . 6  |-  C  =  (  <o  `  K )
354, 29, 11, 30, 34, 51cvrjat 29589 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
3622, 17, 3, 33, 26, 35syl32anc 1192 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( X  .\/  P
)  =  .1.  )
3732, 36breqtrrd 4179 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  .<_  ( X  .\/  P ) )
384, 29, 11, 18, 5cvrat3 29556 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )  ->  (
( Q  =/=  P  /\  -.  P  .<_  X  /\  Q  .<_  ( X  .\/  P ) )  ->  ( X  ./\  ( Q  .\/  P ) )  e.  A
) )
3938imp 419 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )  /\  ( Q  =/=  P  /\  -.  P  .<_  X  /\  Q  .<_  ( X  .\/  P
) ) )  -> 
( X  ./\  ( Q  .\/  P ) )  e.  A )
4022, 23, 25, 26, 37, 39syl23anc 1191 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( X  ./\  ( Q  .\/  P ) )  e.  A )
4121, 40eqeltrd 2461 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   1.cp1 14394   Latclat 14401   OPcops 29287    <o ccvr 29377   Atomscatm 29378   HLchlt 29465
This theorem is referenced by:  cdlemblem  29907  cdlemb  29908  lhpat  30157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
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