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Theorem atcvrj2b 30243
Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
Hypotheses
Ref Expression
atcvrj1x.l  |-  .<_  =  ( le `  K )
atcvrj1x.j  |-  .\/  =  ( join `  K )
atcvrj1x.c  |-  C  =  (  <o  `  K )
atcvrj1x.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvrj2b  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  <-> 
P C ( Q 
.\/  R ) ) )

Proof of Theorem atcvrj2b
StepHypRef Expression
1 simpl3l 1010 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  Q  =/=  R )
21necomd 2542 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R  =/=  Q )
3 simpl1 958 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  K  e.  HL )
4 simpl23 1035 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R  e.  A )
5 simpl22 1034 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  Q  e.  A )
6 atcvrj1x.j . . . . . . . 8  |-  .\/  =  ( join `  K )
7 atcvrj1x.c . . . . . . . 8  |-  C  =  (  <o  `  K )
8 atcvrj1x.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
96, 7, 8atcvr2 30229 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  Q  e.  A )  ->  ( R  =/=  Q  <->  R C ( Q  .\/  R ) ) )
103, 4, 5, 9syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  ( R  =/=  Q  <->  R C
( Q  .\/  R
) ) )
112, 10mpbid 201 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R C ( Q  .\/  R ) )
12 breq1 4042 . . . . . 6  |-  ( P  =  R  ->  ( P C ( Q  .\/  R )  <->  R C ( Q 
.\/  R ) ) )
1312adantl 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  ( P C ( Q  .\/  R )  <->  R C ( Q 
.\/  R ) ) )
1411, 13mpbird 223 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  P C ( Q  .\/  R ) )
15 simpl1 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  K  e.  HL )
16 simpl2 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)
17 simpr 447 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P  =/=  R )
18 simpl3r 1011 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P  .<_  ( Q  .\/  R
) )
19 atcvrj1x.l . . . . . 6  |-  .<_  =  ( le `  K )
2019, 6, 7, 8atcvrj1 30242 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
2115, 16, 17, 18, 20syl112anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P C ( Q  .\/  R ) )
2214, 21pm2.61dane 2537 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
23223expia 1153 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  ->  P C ( Q  .\/  R ) ) )
24 hlatl 30172 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
2524ad2antrr 706 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  AtLat )
26 simplr1 997 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  e.  A )
27 eqid 2296 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 8atn0 30120 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
2925, 26, 28syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  =/=  ( 0. `  K
) )
30 simpll 730 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  HL )
31 eqid 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 8atbase 30101 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3326, 32syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  e.  ( Base `  K
) )
34 simplr2 998 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  e.  A )
35 simplr3 999 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  R  e.  A )
36 simpr 447 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P C ( Q  .\/  R ) )
3731, 6, 27, 7, 8atcvrj0 30239 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  ( Base `  K )  /\  Q  e.  A  /\  R  e.  A )  /\  P C ( Q 
.\/  R ) )  ->  ( P  =  ( 0. `  K
)  <->  Q  =  R
) )
3830, 33, 34, 35, 36, 37syl131anc 1195 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
3938necon3bid 2494 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =/=  ( 0. `  K )  <->  Q  =/=  R ) )
4029, 39mpbid 201 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  =/=  R )
41 hllat 30175 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4241ad2antrr 706 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  Lat )
4331, 8atbase 30101 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4434, 43syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  e.  ( Base `  K
) )
4531, 8atbase 30101 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
4635, 45syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  R  e.  ( Base `  K
) )
4731, 6latjcl 14172 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
4842, 44, 46, 47syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
4930, 33, 483jca 1132 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( K  e.  HL  /\  P  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )
5031, 19, 7cvrle 30090 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5149, 50sylancom 648 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5240, 51jca 518 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R
) ) )
5352ex 423 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P C ( Q  .\/  R )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) ) )
5423, 53impbid 183 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  <-> 
P C ( Q 
.\/  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   0.cp0 14159   Latclat 14167    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  atcvrj2  30244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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