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Theorem cdleme42b 29356
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 5391 . . 3  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2323 . 2  |-  B  e. 
_V
4 nfv 1629 . . 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 3041 . . . . 5  |-  F/_ s [_ R  /  s ]_ N
6 nfcv 2385 . . . . 5  |-  F/_ s  .\/
7 nfcv 2385 . . . . 5  |-  F/_ s
( X  ./\  W
)
85, 6, 7nfov 5733 . . . 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 1631 . . 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 2253 . . . . 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 29269 . . . 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 982 . . 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 3923 . . . . . 6  |-  ( s  =  R  ->  (
s  .<_  W  <->  R  .<_  W ) )
1716notbid 287 . . . . 5  |-  ( s  =  R  ->  ( -.  s  .<_  W  <->  -.  R  .<_  W ) )
18 oveq1 5717 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( R  .\/  ( X  ./\  W ) ) )
1918eqeq1d 2261 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( R  .\/  ( X  ./\  W
) )  =  X ) )
2017, 19anbi12d 694 . . . 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 3017 . . . . . 6  |-  ( s  =  R  ->  N  =  [_ R  /  s ]_ N )
2322oveq1d 5725 . . . . 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 960 . . . 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 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 ) )  ->  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 29318 . . . 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 645 . . 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 1031 . . 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 1032 . . . 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 989 . . . 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 6235 . 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 655 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 939    = wceq 1619    e. wcel 1621   F/_wnfc 2372    =/= wne 2412   A.wral 2509   _Vcvv 2727   [_csb 3009   ifcif 3470   class class class wbr 3920    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme42e  29357  cdleme48fv  29377
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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