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Theorem llni2 33465
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 992 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
2 simpl3 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
3 simpr 461 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  =/=  Q )
4 eqidd 2452 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  =  ( P  .\/  Q ) )
5 neeq1 2729 . . . . 5  |-  ( r  =  P  ->  (
r  =/=  s  <->  P  =/=  s ) )
6 oveq1 6200 . . . . . 6  |-  ( r  =  P  ->  (
r  .\/  s )  =  ( P  .\/  s ) )
76eqeq2d 2465 . . . . 5  |-  ( r  =  P  ->  (
( P  .\/  Q
)  =  ( r 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  s ) ) )
85, 7anbi12d 710 . . . 4  |-  ( r  =  P  ->  (
( r  =/=  s  /\  ( P  .\/  Q
)  =  ( r 
.\/  s ) )  <-> 
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) ) ) )
9 neeq2 2731 . . . . 5  |-  ( s  =  Q  ->  ( P  =/=  s  <->  P  =/=  Q ) )
10 oveq2 6201 . . . . . 6  |-  ( s  =  Q  ->  ( P  .\/  s )  =  ( P  .\/  Q
) )
1110eqeq2d 2465 . . . . 5  |-  ( s  =  Q  ->  (
( P  .\/  Q
)  =  ( P 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  Q ) ) )
129, 11anbi12d 710 . . . 4  |-  ( s  =  Q  ->  (
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) )  <-> 
( P  =/=  Q  /\  ( P  .\/  Q
)  =  ( P 
.\/  Q ) ) ) )
138, 12rspc2ev 3181 . . 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 1223 . 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 991 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
16 eqid 2451 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 llni2.j . . . . 5  |-  .\/  =  ( join `  K )
18 llni2.a . . . . 5  |-  A  =  ( Atoms `  K )
1916, 17, 18hlatjcl 33320 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2019adantr 465 . . 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 33463 . . 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 661 . 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 232 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  N
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   ` cfv 5519  (class class class)co 6193   Basecbs 14285   joincjn 15225   Atomscatm 33217   HLchlt 33304   LLinesclln 33444
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451
This theorem is referenced by:  2atneat  33468  islln2a  33470  2at0mat0  33478  ps-2c  33481  lplnnle2at  33494  2atmat  33514  lplnexllnN  33517  dalempjsen  33606  dalemcea  33613  dalem2  33614  dalemdea  33615  dalem16  33632  dalemcjden  33645  dalem23  33649  dalem54  33679  dalem60  33685  llnexchb2  33822  arglem1N  34143  cdlemc5  34148  cdleme20l1  34273  cdleme20l2  34274  cdleme20l  34275  cdleme22b  34294  cdlemeg46req  34482  cdlemh  34770
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