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Theorem 1cvrat 32493
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 32381 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1018 . . . . 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 1030 . . . . . 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 32307 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 17 . . . . 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 1031 . . . . . 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 32307 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 17 . . . . 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 16013 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
132, 7, 10, 12syl3anc 1230 . . . 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 6293 . . 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 16005 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  P  e.  B )  ->  ( Q  .\/  P
)  e.  B )
162, 10, 7, 15syl3anc 1230 . . . 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 1032 . . . 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 16029 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  P )  e.  B  /\  X  e.  B )  ->  (
( Q  .\/  P
)  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
202, 16, 17, 19syl3anc 1230 . . 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 2443 . 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 997 . . 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 1177 . . 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 1033 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  =/=  Q )
2524necomd 2674 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  =/=  P )
26 simp33 1035 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  -.  P  .<_  X )
27 hlop 32380 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
28273ad2ant1 1018 . . . . 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 32209 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  B )  ->  Q  .<_  .1.  )
3228, 10, 31syl2anc 659 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  .<_  .1.  )
33 simp32 1034 . . . . 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 32492 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
3622, 17, 3, 33, 26, 35syl32anc 1238 . . . 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 4421 . . 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 32459 . . . 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 427 . . 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 1237 . 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 2490 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   1.cp1 15992   Latclat 15999   OPcops 32190    <o ccvr 32280   Atomscatm 32281   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369
This theorem is referenced by:  cdlemblem  32810  cdlemb  32811  lhpat  33060
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