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Theorem lnjatN 29969
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 958 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  K  e.  HL )
2 simpl2 959 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  X  e.  B )
3 simprl 732 . . 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 29968 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
101, 2, 3, 9syl3anc 1182 . 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 983 . . . . 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 1048 . . . . . 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 1049 . . . . . 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 1020 . . . . . 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 1050 . . . . . 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 956 . . . . . 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 2529 . . . . . 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 1021 . . . . . 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 984 . . . . . 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 29967 . . . . . 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 1213 . . . . 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 518 . . . 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 1150 . . 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 2653 . 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 14 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   Linesclines 29683   pmapcpmap 29686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lines 29690  df-pmap 29693
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