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Theorem llni2 32996
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 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 6093 . . . . . 6  |-  ( r  =  P  ->  (
r  .\/  s )  =  ( P  .\/  s ) )
76eqeq2d 2449 . . . . 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 2612 . . . . 5  |-  ( s  =  Q  ->  ( P  =/=  s  <->  P  =/=  Q ) )
10 oveq2 6094 . . . . . 6  |-  ( s  =  Q  ->  ( P  .\/  s )  =  ( P  .\/  Q
) )
1110eqeq2d 2449 . . . . 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 3076 . . 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 1222 . 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 2438 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 llni2.j . . . . 5  |-  .\/  =  ( join `  K )
18 llni2.a . . . . 5  |-  A  =  ( Atoms `  K )
1916, 17, 18hlatjcl 32851 . . . 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 32994 . . 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 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   ` cfv 5413  (class class class)co 6086   Basecbs 14166   joincjn 15106   Atomscatm 32748   HLchlt 32835   LLinesclln 32975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982
This theorem is referenced by:  2atneat  32999  islln2a  33001  2at0mat0  33009  ps-2c  33012  lplnnle2at  33025  2atmat  33045  lplnexllnN  33048  dalempjsen  33137  dalemcea  33144  dalem2  33145  dalemdea  33146  dalem16  33163  dalemcjden  33176  dalem23  33180  dalem54  33210  dalem60  33216  llnexchb2  33353  arglem1N  33674  cdlemc5  33679  cdleme20l1  33804  cdleme20l2  33805  cdleme20l  33806  cdleme22b  33825  cdlemeg46req  34013  cdlemh  34301
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