Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lnjatN Structured version   Unicode version

Theorem lnjatN 33727
Description: Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnjat.b  |-  B  =  ( Base `  K
)
lnjat.l  |-  .<_  =  ( le `  K )
lnjat.j  |-  .\/  =  ( join `  K )
lnjat.a  |-  A  =  ( Atoms `  K )
lnjat.n  |-  N  =  ( Lines `  K )
lnjat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
lnjatN  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  X  =  ( P  .\/  q
) ) )
Distinct variable groups:    A, q    B, q    K, q    .<_ , q    M, q    N, q    P, q    X, q
Allowed substitution hint:    .\/ ( q)

Proof of Theorem lnjatN
StepHypRef Expression
1 simpl1 991 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl2 992 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 755 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( M `  X )  e.  N
)
4 lnjat.b . . . 4  |-  B  =  ( Base `  K
)
5 lnjat.l . . . 4  |-  .<_  =  ( le `  K )
6 lnjat.a . . . 4  |-  A  =  ( Atoms `  K )
7 lnjat.n . . . 4  |-  N  =  ( Lines `  K )
8 lnjat.m . . . 4  |-  M  =  ( pmap `  K
)
94, 5, 6, 7, 8lnatexN 33726 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
101, 2, 3, 9syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
11 simp3l 1016 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  q  =/=  P )
12 simp1l1 1081 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  K  e.  HL )
13 simp1l2 1082 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  X  e.  B )
14 simp1rl 1053 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  ( M `  X )  e.  N
)
15 simp1l3 1083 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  P  e.  A )
16 simp2 989 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  q  e.  A )
1711necomd 2717 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  P  =/=  q )
18 simp1rr 1054 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  P  .<_  X )
19 simp3r 1017 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  q  .<_  X )
20 lnjat.j . . . . . . 7  |-  .\/  =  ( join `  K )
214, 5, 20, 6, 7, 8lneq2at 33725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  q  e.  A  /\  P  =/=  q
)  /\  ( P  .<_  X  /\  q  .<_  X ) )  ->  X  =  ( P  .\/  q ) )
2212, 13, 14, 15, 16, 17, 18, 19, 21syl332anc 1250 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  X  =  ( P  .\/  q ) )
2311, 22jca 532 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  .<_  X ) )  ->  ( q  =/=  P  /\  X  =  ( P  .\/  q
) ) )
24233exp 1187 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( q  e.  A  ->  ( ( q  =/=  P  /\  q  .<_  X )  -> 
( q  =/=  P  /\  X  =  ( P  .\/  q ) ) ) ) )
2524reximdvai 2919 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( E. q  e.  A  (
q  =/=  P  /\  q  .<_  X )  ->  E. q  e.  A  ( q  =/=  P  /\  X  =  ( P  .\/  q ) ) ) )
2610, 25mpd 15 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  E. q  e.  A  ( q  =/=  P  /\  X  =  ( P  .\/  q
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   E.wrex 2794   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   joincjn 15213   Atomscatm 33211   HLchlt 33298   Linesclines 33441   pmapcpmap 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 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-lines 33448  df-pmap 33451
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator