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Theorem cdleme40w 30732
Description: Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 30731 bound variable change to  [_ S  /  u ]_ V. TODO: FIX COMMENT (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme40.b  |-  B  =  ( Base `  K
)
cdleme40.l  |-  .<_  =  ( le `  K )
cdleme40.j  |-  .\/  =  ( join `  K )
cdleme40.m  |-  ./\  =  ( meet `  K )
cdleme40.a  |-  A  =  ( Atoms `  K )
cdleme40.h  |-  H  =  ( LHyp `  K
)
cdleme40.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme40.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme40.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme40.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme40.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme40.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme40r.y  |-  Y  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
Assertion
Ref Expression
cdleme40w  |-  ( ( ( ( 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:    u, A    u, B    u,  .\/    u,  .<_    u,  ./\    u, P    u, Q    u, S    u, W    t, s,
y, A    B, s,
t, y    E, s    t, H, y    .\/ , s,
t, y    t, K, y   
.<_ , s, t, y    ./\ , s,
t, y    P, s,
t, y    Q, s,
t, y    R, s,
t, y    t, U, y    W, s, t, y   
y, Y    t, S, y    y, E    u, N    S, s, u    U, s, u, t, y
Allowed substitution hints:    D( y, u, t, s)    R( u)    E( u, t)    G( y, u, t, s)    H( u, s)    I( y, u, t, s)    K( u, s)    N( y, t, s)    Y( u, t, s)

Proof of Theorem cdleme40w
Dummy variables  v 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdleme40.b . . 3  |-  B  =  ( Base `  K
)
2 cdleme40.l . . 3  |-  .<_  =  ( le `  K )
3 cdleme40.j . . 3  |-  .\/  =  ( join `  K )
4 cdleme40.m . . 3  |-  ./\  =  ( meet `  K )
5 cdleme40.a . . 3  |-  A  =  ( Atoms `  K )
6 cdleme40.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdleme40.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdleme40.e . . 3  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdleme40.g . . 3  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 cdleme40.i . . 3  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
11 cdleme40.n . . 3  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
12 eqid 2285 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
13 eqid 2285 . . 3  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) ) ) )  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) ) ) )
14 eqid 2285 . . 3  |-  ( ( v  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  =  ( ( v  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )
15 eqid 2285 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( ( ( v  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( S  .\/  v )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( S  .\/  v )  ./\  W
) ) )
16 eqid 2285 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( ( ( v  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) )
17 eqid 2285 . . 3  |-  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) ) ) )  =  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) ) ) )
18 eqid 2285 . . 3  |-  if ( u  .<_  ( P  .\/  Q ) ,  (
iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  ( ( P 
.\/  Q )  ./\  ( ( ( v 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v ) 
./\  W ) ) )  .\/  ( ( u  .\/  v ) 
./\  W ) ) ) ) ) ,  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) ) )  =  if ( u 
.<_  ( P  .\/  Q
) ,  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) ) ) ) ,  ( ( u  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) ) )
19 eqid 2285 . . 3  |-  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( S  .\/  v )  ./\  W
) ) ) ) )  =  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( S  .\/  v )  ./\  W
) ) ) ) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme40n 30730 . 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 ) )  ->  [_ R  /  s ]_ N  =/=  [_ S  /  u ]_ if ( u  .<_  ( P  .\/  Q ) ,  (
iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  ( ( P 
.\/  Q )  ./\  ( ( ( v 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v ) 
./\  W ) ) )  .\/  ( ( u  .\/  v ) 
./\  W ) ) ) ) ) ,  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) ) ) )
21 simp23l 1076 . . 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 )
22 cdleme40.d . . . 4  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
23 eqid 2285 . . . 4  |-  ( ( u  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 14, 16, 17, 18cdleme40v 30731 . . 3  |-  ( S  e.  A  ->  [_ S  /  s ]_ N  =  [_ S  /  u ]_ if ( u  .<_  ( P  .\/  Q ) ,  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( (
( v  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) ) ) ) ,  ( ( u  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) ) ) )
2521, 24syl 15 . 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  =  [_ S  /  u ]_ if ( u  .<_  ( P  .\/  Q ) ,  (
iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  ( ( P 
.\/  Q )  ./\  ( ( ( v 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v ) 
./\  W ) ) )  .\/  ( ( u  .\/  v ) 
./\  W ) ) ) ) ) ,  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) ) ) )
2620, 25neeqtrrd 2472 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   [_csb 3083   ifcif 3567   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   iota_crio 6299   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   Atomscatm 29526   HLchlt 29613   LHypclh 30246
This theorem is referenced by:  cdleme41snaw  30738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250
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