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Theorem cdleme40w 29909
Description: Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 29908 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
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 2258 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
13 eqid 2258 . . 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 2258 . . 3  |-  ( ( v  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )  =  ( ( v  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )
15 eqid 2258 . . 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 2258 . . 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 2258 . . 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 2258 . . 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 2258 . . 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 29907 . 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 1081 . . 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 2258 . . . 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 29908 . . 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 17 . 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 2445 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   [_csb 3056   ifcif 3539   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Atomscatm 28703   HLchlt 28790   LHypclh 29423
This theorem is referenced by:  cdleme41snaw  29915
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427
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