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Theorem lnjatN 35920
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 997 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl2 998 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 754 . . 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 35919 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
101, 2, 3, 9syl3anc 1226 . 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 1022 . . . . 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 1087 . . . . . 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 1088 . . . . . 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 1059 . . . . . 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 1089 . . . . . 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 995 . . . . . 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 2725 . . . . . 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 1060 . . . . . 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 1023 . . . . . 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 35918 . . . . . 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 1257 . . . . 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 530 . . . 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 1193 . . 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 2926 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   Atomscatm 35404   HLchlt 35491   Linesclines 35634   pmapcpmap 35637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-lines 35641  df-pmap 35644
This theorem is referenced by: (None)
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