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Theorem llni2 33148
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 1034 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
2 simpl3 1035 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
3 simpr 468 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  =/=  Q )
4 eqidd 2472 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  =  ( P  .\/  Q ) )
5 neeq1 2705 . . . . 5  |-  ( r  =  P  ->  (
r  =/=  s  <->  P  =/=  s ) )
6 oveq1 6315 . . . . . 6  |-  ( r  =  P  ->  (
r  .\/  s )  =  ( P  .\/  s ) )
76eqeq2d 2481 . . . . 5  |-  ( r  =  P  ->  (
( P  .\/  Q
)  =  ( r 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  s ) ) )
85, 7anbi12d 725 . . . 4  |-  ( r  =  P  ->  (
( r  =/=  s  /\  ( P  .\/  Q
)  =  ( r 
.\/  s ) )  <-> 
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) ) ) )
9 neeq2 2706 . . . . 5  |-  ( s  =  Q  ->  ( P  =/=  s  <->  P  =/=  Q ) )
10 oveq2 6316 . . . . . 6  |-  ( s  =  Q  ->  ( P  .\/  s )  =  ( P  .\/  Q
) )
1110eqeq2d 2481 . . . . 5  |-  ( s  =  Q  ->  (
( P  .\/  Q
)  =  ( P 
.\/  s )  <->  ( P  .\/  Q )  =  ( P  .\/  Q ) ) )
129, 11anbi12d 725 . . . 4  |-  ( s  =  Q  ->  (
( P  =/=  s  /\  ( P  .\/  Q
)  =  ( P 
.\/  s ) )  <-> 
( P  =/=  Q  /\  ( P  .\/  Q
)  =  ( P 
.\/  Q ) ) ) )
138, 12rspc2ev 3149 . . 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 1296 . 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 1033 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
16 eqid 2471 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 llni2.j . . . . 5  |-  .\/  =  ( join `  K )
18 llni2.a . . . . 5  |-  A  =  ( Atoms `  K )
1916, 17, 18hlatjcl 33003 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2019adantr 472 . . 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 33146 . . 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 673 . 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 240 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   ` cfv 5589  (class class class)co 6308   Basecbs 15199   joincjn 16267   Atomscatm 32900   HLchlt 32987   LLinesclln 33127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134
This theorem is referenced by:  2atneat  33151  islln2a  33153  2at0mat0  33161  ps-2c  33164  lplnnle2at  33177  2atmat  33197  lplnexllnN  33200  dalempjsen  33289  dalemcea  33296  dalem2  33297  dalemdea  33298  dalem16  33315  dalemcjden  33328  dalem23  33332  dalem54  33362  dalem60  33368  llnexchb2  33505  arglem1N  33827  cdlemc5  33832  cdleme20l1  33958  cdleme20l2  33959  cdleme20l  33960  cdleme22b  33979  cdlemeg46req  34167  cdlemh  34455
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