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Theorem 1cvrat 33439
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 33327 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . . . 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 1021 . . . . . 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 33253 . . . . . 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 1022 . . . . . 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 33253 . . . . . 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 15343 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
132, 7, 10, 12syl3anc 1219 . . . 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 6210 . . 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 15335 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  P  e.  B )  ->  ( Q  .\/  P
)  e.  B )
162, 10, 7, 15syl3anc 1219 . . . 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 1023 . . . 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 15359 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  P )  e.  B  /\  X  e.  B )  ->  (
( Q  .\/  P
)  ./\  X )  =  ( X  ./\  ( Q  .\/  P ) ) )
202, 16, 17, 19syl3anc 1219 . . 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 2493 . 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 988 . . 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 1168 . . 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 1024 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  P  =/=  Q )
2524necomd 2720 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  =/=  P )
26 simp33 1026 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  -.  P  .<_  X )
27 hlop 33326 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
28273ad2ant1 1009 . . . . 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 33155 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  B )  ->  Q  .<_  .1.  )
3228, 10, 31syl2anc 661 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  ->  Q  .<_  .1.  )
33 simp32 1025 . . . . 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 33438 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
3622, 17, 3, 33, 26, 35syl32anc 1227 . . . 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 33405 . . . 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 429 . . 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 1226 . 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 2540 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   1.cp1 15322   Latclat 15329   OPcops 33136    <o ccvr 33226   Atomscatm 33227   HLchlt 33314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315
This theorem is referenced by:  cdlemblem  33756  cdlemb  33757  lhpat  34006
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