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Theorem cdleme42b 29935
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 5500 . . 3  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2355 . 2  |-  B  e. 
_V
4 nfv 1606 . . 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 3115 . . . . 5  |-  F/_ s [_ R  /  s ]_ N
6 nfcv 2421 . . . . 5  |-  F/_ s  .\/
7 nfcv 2421 . . . . 5  |-  F/_ s
( X  ./\  W
)
85, 6, 7nfov 5843 . . . 4  |-  F/_ s
( [_ R  /  s ]_ N  .\/  ( X 
./\  W ) )
98a1i 12 . . 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 1607 . . 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 2285 . . . . 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 29848 . . . 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 4028 . . . . . 6  |-  ( s  =  R  ->  (
s  .<_  W  <->  R  .<_  W ) )
1716notbid 287 . . . . 5  |-  ( s  =  R  ->  ( -.  s  .<_  W  <->  -.  R  .<_  W ) )
18 oveq1 5827 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( R  .\/  ( X  ./\  W ) ) )
1918eqeq1d 2293 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( R  .\/  ( X  ./\  W
) )  =  X ) )
2017, 19anbi12d 693 . . . 4  |-  ( s  =  R  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) ) )
2120adantl 454 . . 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 3091 . . . . . 6  |-  ( s  =  R  ->  N  =  [_ R  /  s ]_ N )
2322oveq1d 5835 . . . . 5  |-  ( s  =  R  ->  ( N  .\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
2423a1i 12 . . . 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 ) )  ->  ( s  =  R  ->  ( N 
.\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) ) )
2524imp 420 . . 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 ) ) )
26 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 ) ) )
27 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 )
28 cdleme41.l . . . . 5  |-  .<_  =  ( le `  K )
29 cdleme41.j . . . . 5  |-  .\/  =  ( join `  K )
30 cdleme41.m . . . . 5  |-  ./\  =  ( meet `  K )
31 cdleme41.a . . . . 5  |-  A  =  ( Atoms `  K )
32 cdleme41.h . . . . 5  |-  H  =  ( LHyp `  K
)
33 cdleme41.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
34 cdleme41.d . . . . 5  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
35 cdleme41.e . . . . 5  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
36 cdleme41.g . . . . 5  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
37 cdleme41.i . . . . 5  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
38 cdleme41.n . . . . 5  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
391, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 11, 12cdleme32fvcl 29897 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( F `  X
)  e.  B )
4026, 27, 39syl2anc 644 . . 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
)
41 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 )
42 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 )
43 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 )
4442, 43jca 520 . . 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 ) )
454, 9, 10, 15, 21, 25, 40, 41, 44riotasv2d 6345 . 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
) ) )
463, 45mpan2 654 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   F/_wnfc 2408    =/= wne 2448   A.wral 2545   _Vcvv 2790   [_csb 3083   ifcif 3567   class class class wbr 4025    e. cmpt 4079   ` cfv 5222  (class class class)co 5820   iota_crio 6291   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Atomscatm 28721   HLchlt 28808   LHypclh 29441
This theorem is referenced by:  cdleme42e  29936  cdleme48fv  29956
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445
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