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Theorem atcvrj2b 29621
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 2529 . . . . . 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 29607 . . . . . . 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 4026 . . . . . 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 29620 . . . . 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 2524 . . 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 29550 . . . . . . 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 2283 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 8atn0 29498 . . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 8atbase 29479 . . . . . . . 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 29617 . . . . . . 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 2481 . . . . 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 29553 . . . . . . . 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 29479 . . . . . . . 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 29479 . . . . . . . 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 14156 . . . . . . 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 29468 . . . . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   0.cp0 14143   Latclat 14151    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  atcvrj2  29622
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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