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Theorem llni2 30323
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 959 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
2 simpl3 960 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
3 simpr 447 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  =/=  Q )
4 eqidd 2297 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  =  ( P  .\/  Q ) )
5 neeq1 2467 . . . . 5  |-  ( r  =  P  ->  (
r  =/=  s  <->  P  =/=  s ) )
6 oveq1 5881 . . . . . 6  |-  ( r  =  P  ->  (
r  .\/  s )  =  ( P  .\/  s ) )
76eqeq2d 2307 . . . . 5  |-  ( r  =  P  ->  (
( P  .\/  Q
)  =  ( r 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  s ) ) )
85, 7anbi12d 691 . . . 4  |-  ( r  =  P  ->  (
( r  =/=  s  /\  ( P  .\/  Q
)  =  ( r 
.\/  s ) )  <-> 
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) ) ) )
9 neeq2 2468 . . . . 5  |-  ( s  =  Q  ->  ( P  =/=  s  <->  P  =/=  Q ) )
10 oveq2 5882 . . . . . 6  |-  ( s  =  Q  ->  ( P  .\/  s )  =  ( P  .\/  Q
) )
1110eqeq2d 2307 . . . . 5  |-  ( s  =  Q  ->  (
( P  .\/  Q
)  =  ( P 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  Q ) ) )
129, 11anbi12d 691 . . . 4  |-  ( s  =  Q  ->  (
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) )  <-> 
( P  =/=  Q  /\  ( P  .\/  Q
)  =  ( P 
.\/  Q ) ) ) )
138, 12rspc2ev 2905 . . 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 1186 . 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 958 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
16 eqid 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 llni2.j . . . . 5  |-  .\/  =  ( join `  K )
18 llni2.a . . . . 5  |-  A  =  ( Atoms `  K )
1916, 17, 18hlatjcl 30178 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2019adantr 451 . . 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 30321 . . 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 642 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   Atomscatm 30075   HLchlt 30162   LLinesclln 30302
This theorem is referenced by:  2atneat  30326  islln2a  30328  2at0mat0  30336  ps-2c  30339  lplnnle2at  30352  2atmat  30372  lplnexllnN  30375  dalempjsen  30464  dalemcea  30471  dalem2  30472  dalemdea  30473  dalem16  30490  dalemcjden  30503  dalem23  30507  dalem54  30537  dalem60  30543  llnexchb2  30680  arglem1N  31001  cdlemc5  31006  cdleme20l1  31131  cdleme20l2  31132  cdleme20l  31133  cdleme22b  31152  cdlemeg46req  31340  cdlemh  31628
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  df-llines 30309
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