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Theorem cdleme42b 30960
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 6-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
)
cdleme41.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme41.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme42b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, 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, U, y   
t, W, y    x, z, A    x, B, z   
z, E, s    z, H    x,  .\/ , z    z, K   
x,  .<_ , z    x,  ./\ , z    x, N, z    x, P, z    x, Q, z   
x, R, z    x, U, z    x, W, z, s, t, y    X, s, t, x, z
Allowed substitution hints:    D( x, z, t, s)    E( x, t)    F( x, y, z, t, s)    G( x, z, t, s)    H( x)    I( x, y, z, t, s)    K( x)    N( y, t, s)    O( x, y, z, t, s)    X( y)

Proof of Theorem cdleme42b
StepHypRef Expression
1 cdleme41.b . . 3  |-  B  =  ( Base `  K
)
2 fvex 5701 . . 3  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2474 . 2  |-  B  e. 
_V
4 nfv 1626 . . 3  |-  F/ s ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )
5 nfcsb1v 3243 . . . . 5  |-  F/_ s [_ R  /  s ]_ N
6 nfcv 2540 . . . . 5  |-  F/_ s  .\/
7 nfcv 2540 . . . . 5  |-  F/_ s
( X  ./\  W
)
85, 6, 7nfov 6063 . . . 4  |-  F/_ s
( [_ R  /  s ]_ N  .\/  ( X 
./\  W ) )
98a1i 11 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  F/_ s (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
10 nfvd 1627 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  F/ s
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )
11 cdleme41.o . . . . 5  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
12 cdleme41.f . . . . 5  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
13 eqid 2404 . . . . 5  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )
1411, 12, 13cdleme31fv1 30873 . . . 4  |-  ( ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  (
iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
15143ad2ant2 979 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  (
iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
16 breq1 4175 . . . . . 6  |-  ( s  =  R  ->  (
s  .<_  W  <->  R  .<_  W ) )
1716notbid 286 . . . . 5  |-  ( s  =  R  ->  ( -.  s  .<_  W  <->  -.  R  .<_  W ) )
18 oveq1 6047 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( R  .\/  ( X  ./\  W ) ) )
1918eqeq1d 2412 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( R  .\/  ( X  ./\  W
) )  =  X ) )
2017, 19anbi12d 692 . . . 4  |-  ( s  =  R  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) ) )
2120adantl 453 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  /\  s  =  R )  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) ) )
22 csbeq1a 3219 . . . . 5  |-  ( s  =  R  ->  N  =  [_ R  /  s ]_ N )
2322oveq1d 6055 . . . 4  |-  ( s  =  R  ->  ( N  .\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
2423adantl 453 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  /\  s  =  R )  ->  ( N  .\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
25 simp1 957 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
26 simp2l 983 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
27 cdleme41.l . . . . 5  |-  .<_  =  ( le `  K )
28 cdleme41.j . . . . 5  |-  .\/  =  ( join `  K )
29 cdleme41.m . . . . 5  |-  ./\  =  ( meet `  K )
30 cdleme41.a . . . . 5  |-  A  =  ( Atoms `  K )
31 cdleme41.h . . . . 5  |-  H  =  ( LHyp `  K
)
32 cdleme41.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
33 cdleme41.d . . . . 5  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
34 cdleme41.e . . . . 5  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
35 cdleme41.g . . . . 5  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
36 cdleme41.i . . . . 5  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
37 cdleme41.n . . . . 5  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
381, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 11, 12cdleme32fvcl 30922 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( F `  X
)  e.  B )
3925, 26, 38syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  e.  B
)
40 simp3ll 1028 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  R  e.  A )
41 simp3lr 1029 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  R  .<_  W )
42 simp3r 986 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( R  .\/  ( X  ./\  W
) )  =  X )
4341, 42jca 519 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )
444, 9, 10, 15, 21, 24, 39, 40, 43riotasv2d 6553 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  /\  B  e. 
_V )  ->  ( F `  X )  =  ( [_ R  /  s ]_ N  .\/  ( X  ./\  W
) ) )
453, 44mpan2 653 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   F/_wnfc 2527    =/= wne 2567   A.wral 2666   _Vcvv 2916   [_csb 3211   ifcif 3699   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme42e  30961  cdleme48fv  30981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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