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Theorem 1cvrat 28944
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 28832 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . . . 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 988 . . . . . 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 28758 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 15 . . . . 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 989 . . . . . 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 28758 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 15 . . . . 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 14161 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
132, 7, 10, 12syl3anc 1182 . . . 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 5835 . . 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 14152 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  P  e.  B )  ->  ( Q  .\/  P
)  e.  B )
162, 10, 7, 15syl3anc 1182 . . . 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 990 . . . 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 14177 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  P )  e.  B  /\  X  e.  B )  ->  (
( Q  .\/  P
)  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
202, 16, 17, 19syl3anc 1182 . . 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 2316 . 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 955 . . 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 1132 . . 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 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  =/=  Q )
2524necomd 2530 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  =/=  P )
26 simp33 993 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  -.  P  .<_  X )
27 hlop 28831 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
28273ad2ant1 976 . . . . 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 28660 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  B )  ->  Q  .<_  .1.  )
3228, 10, 31syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  .<_  .1.  )
33 simp32 992 . . . . 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 28943 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
3622, 17, 3, 33, 26, 35syl32anc 1190 . . . 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 4050 . . 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 28910 . . . 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 418 . . 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 1189 . 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 2358 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 358    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   1.cp1 14140   Latclat 14147   OPcops 28641    <o ccvr 28731   Atomscatm 28732   HLchlt 28819
This theorem is referenced by:  cdlemblem  29261  cdlemb  29262  lhpat  29511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820
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