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Theorem lnjatN 33044
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 1008 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl2 1009 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 762 . . 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 33043 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
101, 2, 3, 9syl3anc 1264 . 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 1033 . . . . 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 1098 . . . . . 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 1099 . . . . . 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 1070 . . . . . 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 1100 . . . . . 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 1006 . . . . . 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 2702 . . . . . 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 1071 . . . . . 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 1034 . . . . . 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 33042 . . . . . 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 1295 . . . . 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 534 . . . 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 1204 . . 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 2904 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15075   lecple 15150   joincjn 16131   Atomscatm 32528   HLchlt 32615   Linesclines 32758   pmapcpmap 32761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16115  df-poset 16133  df-plt 16146  df-lub 16162  df-glb 16163  df-join 16164  df-meet 16165  df-p0 16227  df-lat 16234  df-clat 16296  df-oposet 32441  df-ol 32443  df-oml 32444  df-covers 32531  df-ats 32532  df-atl 32563  df-cvlat 32587  df-hlat 32616  df-lines 32765  df-pmap 32768
This theorem is referenced by: (None)
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