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Theorem llni2 30371
Description: The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)
Hypotheses
Ref Expression
llni2.j  |-  .\/  =  ( join `  K )
llni2.a  |-  A  =  ( Atoms `  K )
llni2.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llni2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)

Proof of Theorem llni2
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 962 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
2 simpl3 963 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
3 simpr 449 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  =/=  Q )
4 eqidd 2439 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  =  ( P  .\/  Q ) )
5 neeq1 2611 . . . . 5  |-  ( r  =  P  ->  (
r  =/=  s  <->  P  =/=  s ) )
6 oveq1 6090 . . . . . 6  |-  ( r  =  P  ->  (
r  .\/  s )  =  ( P  .\/  s ) )
76eqeq2d 2449 . . . . 5  |-  ( r  =  P  ->  (
( P  .\/  Q
)  =  ( r 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  s ) ) )
85, 7anbi12d 693 . . . 4  |-  ( r  =  P  ->  (
( r  =/=  s  /\  ( P  .\/  Q
)  =  ( r 
.\/  s ) )  <-> 
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) ) ) )
9 neeq2 2612 . . . . 5  |-  ( s  =  Q  ->  ( P  =/=  s  <->  P  =/=  Q ) )
10 oveq2 6091 . . . . . 6  |-  ( s  =  Q  ->  ( P  .\/  s )  =  ( P  .\/  Q
) )
1110eqeq2d 2449 . . . . 5  |-  ( s  =  Q  ->  (
( P  .\/  Q
)  =  ( P 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  Q ) ) )
129, 11anbi12d 693 . . . 4  |-  ( s  =  Q  ->  (
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) )  <-> 
( P  =/=  Q  /\  ( P  .\/  Q
)  =  ( P 
.\/  Q ) ) ) )
138, 12rspc2ev 3062 . . 3  |-  ( ( P  e.  A  /\  Q  e.  A  /\  ( P  =/=  Q  /\  ( P  .\/  Q
)  =  ( P 
.\/  Q ) ) )  ->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  ( P  .\/  Q )  =  ( r  .\/  s
) ) )
141, 2, 3, 4, 13syl112anc 1189 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  ( P  .\/  Q )  =  ( r  .\/  s
) ) )
15 simpl1 961 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
16 eqid 2438 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 llni2.j . . . . 5  |-  .\/  =  ( join `  K )
18 llni2.a . . . . 5  |-  A  =  ( Atoms `  K )
1916, 17, 18hlatjcl 30226 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2019adantr 453 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
21 llni2.n . . . 4  |-  N  =  ( LLines `  K )
2216, 17, 18, 21islln3 30369 . . 3  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  e.  N  <->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  ( P  .\/  Q )  =  ( r  .\/  s
) ) ) )
2315, 20, 22syl2anc 644 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  Q )  e.  N  <->  E. r  e.  A  E. s  e.  A  ( r  =/=  s  /\  ( P  .\/  Q
)  =  ( r 
.\/  s ) ) ) )
2414, 23mpbird 225 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   ` cfv 5456  (class class class)co 6083   Basecbs 13471   joincjn 14403   Atomscatm 30123   HLchlt 30210   LLinesclln 30350
This theorem is referenced by:  2atneat  30374  islln2a  30376  2at0mat0  30384  ps-2c  30387  lplnnle2at  30400  2atmat  30420  lplnexllnN  30423  dalempjsen  30512  dalemcea  30519  dalem2  30520  dalemdea  30521  dalem16  30538  dalemcjden  30551  dalem23  30555  dalem54  30585  dalem60  30591  llnexchb2  30728  arglem1N  31049  cdlemc5  31054  cdleme20l1  31179  cdleme20l2  31180  cdleme20l  31181  cdleme22b  31200  cdlemeg46req  31388  cdlemh  31676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357
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