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Theorem 1cvrat 30371
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 30259 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 979 . . . . 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 991 . . . . . 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 30185 . . . . . 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 992 . . . . . 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 30185 . . . . . 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 14519 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
132, 7, 10, 12syl3anc 1185 . . . 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 6125 . . 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 14510 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  P  e.  B )  ->  ( Q  .\/  P
)  e.  B )
162, 10, 7, 15syl3anc 1185 . . . 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 993 . . . 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 14535 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  P )  e.  B  /\  X  e.  B )  ->  (
( Q  .\/  P
)  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
202, 16, 17, 19syl3anc 1185 . . 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 2474 . 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 958 . . 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 1135 . . 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 994 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  =/=  Q )
2524necomd 2693 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  =/=  P )
26 simp33 996 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  -.  P  .<_  X )
27 hlop 30258 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
28273ad2ant1 979 . . . . 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 30087 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  B )  ->  Q  .<_  .1.  )
3228, 10, 31syl2anc 644 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  .<_  .1.  )
33 simp32 995 . . . . 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 30370 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
3622, 17, 3, 33, 26, 35syl32anc 1193 . . . 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 4263 . . 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 30337 . . . 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 420 . . 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 1192 . 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 2516 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   joincjn 14432   meetcmee 14433   1.cp1 14498   Latclat 14505   OPcops 30068    <o ccvr 30158   Atomscatm 30159   HLchlt 30246
This theorem is referenced by:  cdlemblem  30688  cdlemb  30689  lhpat  30938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247
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