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

Theorem atbtwnexOLDN 32931
Description: There exists a 3rd atom  r on a line  P  .\/  Q intersecting element  X at  P, such that  r is different from  Q and not in  X. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atbtwn.b  |-  B  =  ( Base `  K
)
atbtwn.l  |-  .<_  =  ( le `  K )
atbtwn.j  |-  .\/  =  ( join `  K )
atbtwn.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atbtwnexOLDN  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    B, r    K, r    .<_ , r    P, r    Q, r    X, r
Allowed substitution hint:    .\/ ( r)

Proof of Theorem atbtwnexOLDN
StepHypRef Expression
1 simpr2 1012 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
2 simpr3 1013 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
3 nbrne2 4439 . . . 4  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
41, 2, 3syl2anc 665 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
5 atbtwn.l . . . 4  |-  .<_  =  ( le `  K )
6 atbtwn.j . . . 4  |-  .\/  =  ( join `  K )
7 atbtwn.a . . . 4  |-  A  =  ( Atoms `  K )
85, 6, 7hlsupr 32870 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
94, 8syldan 472 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
10 simp32 1042 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  =/=  Q )
11 simp31 1041 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  =/=  P )
12 simp1l 1029 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
13 simp2 1006 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  e.  A )
14 simp1r1 1101 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  X  e.  B )
15 simp1r2 1102 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  X )
16 simp1r3 1103 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  -.  Q  .<_  X )
17 simp33 1043 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  .<_  ( P  .\/  Q ) )
18 atbtwn.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1918, 5, 6, 7atbtwn 32930 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( r  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )  ->  (
r  =/=  P  <->  -.  r  .<_  X ) )
2012, 13, 14, 15, 16, 17, 19syl123anc 1281 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  =/=  P  <->  -.  r  .<_  X ) )
2111, 20mpbid 213 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  -.  r  .<_  X )
2210, 21, 173jca 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
23223exp 1204 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  ( r  e.  A  ->  ( ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q
) )  ->  (
r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) ) ) )
2423reximdvai 2897 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  ( E. r  e.  A  (
r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q
) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) ) )
259, 24mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776   class class class wbr 4420   ` cfv 5598  (class class class)co 6302   Basecbs 15109   lecple 15185   joincjn 16177   Atomscatm 32748   HLchlt 32835
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-preset 16161  df-poset 16179  df-plt 16192  df-lub 16208  df-glb 16209  df-join 16210  df-meet 16211  df-p0 16273  df-lat 16280  df-clat 16342  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator