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Theorem cdleme41sn4aw 36617
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for on and off  P  .\/  Q line. TODO: FIX COMMENT (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme41.b  |-  B  =  ( Base `  K
)
cdleme41.l  |-  .<_  =  ( le `  K )
cdleme41.j  |-  .\/  =  ( join `  K )
cdleme41.m  |-  ./\  =  ( meet `  K )
cdleme41.a  |-  A  =  ( Atoms `  K )
cdleme41.h  |-  H  =  ( LHyp `  K
)
cdleme41.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme41.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme41.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme41.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme41.i  |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme41.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
Assertion
Ref Expression
cdleme41sn4aw  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  [_ R  /  s ]_ N  =/=  [_ S  /  s ]_ N
)
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, s    S, s    U, s    W, s    y, t, A, s    B, s, t, y    y, D    y, G    E, s,
y    H, s, t, y   
t,  .\/ , y    K, s, t, y    t,  .<_ , y   
t,  ./\ , y    t, P, y    t, Q, y    t, R, y    t, S, y   
t, U, y    t, W, y
Allowed substitution hints:    D( t, s)    E( t)    G( t, s)    I( y, t, s)    N( y, t, s)

Proof of Theorem cdleme41sn4aw
StepHypRef Expression
1 simp1 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp21 1027 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  P  =/=  Q
)
3 simp23 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
4 simp22 1028 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5 simp32 1031 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  S  .<_  ( P 
.\/  Q ) )
6 simp31 1030 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
7 simp33 1032 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  R  =/=  S
)
87necomd 2725 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  S  =/=  R
)
9 cdleme41.b . . . 4  |-  B  =  ( Base `  K
)
10 cdleme41.l . . . 4  |-  .<_  =  ( le `  K )
11 cdleme41.j . . . 4  |-  .\/  =  ( join `  K )
12 cdleme41.m . . . 4  |-  ./\  =  ( meet `  K )
13 cdleme41.a . . . 4  |-  A  =  ( Atoms `  K )
14 cdleme41.h . . . 4  |-  H  =  ( LHyp `  K
)
15 cdleme41.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
16 cdleme41.d . . . 4  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
17 cdleme41.e . . . 4  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
18 cdleme41.g . . . 4  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
19 cdleme41.i . . . 4  |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
20 cdleme41.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
219, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20cdleme41sn3aw 36616 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  .<_  ( P 
.\/  Q )  /\  -.  R  .<_  ( P 
.\/  Q )  /\  S  =/=  R ) )  ->  [_ S  /  s ]_ N  =/=  [_ R  /  s ]_ N
)
221, 2, 3, 4, 5, 6, 8, 21syl133anc 1249 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  [_ S  /  s ]_ N  =/=  [_ R  /  s ]_ N
)
2322necomd 2725 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  R  =/=  S ) )  ->  [_ R  /  s ]_ N  =/=  [_ S  /  s ]_ N
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   [_csb 3420   ifcif 3929   class class class wbr 4439   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124
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  ax-riotaBAD 35100
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  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-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  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-iin 4318  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-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  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-p1 15872  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-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128
This theorem is referenced by:  cdleme41snaw  36618
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