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Theorem cdleme32a 33440
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-2013.)
Hypotheses
Ref Expression
cdleme32.b  |-  B  =  ( Base `  K
)
cdleme32.l  |-  .<_  =  ( le `  K )
cdleme32.j  |-  .\/  =  ( join `  K )
cdleme32.m  |-  ./\  =  ( meet `  K )
cdleme32.a  |-  A  =  ( Atoms `  K )
cdleme32.h  |-  H  =  ( LHyp `  K
)
cdleme32.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme32.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme32.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme32.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme32.i  |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
cdleme32.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
cdleme32.o  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme32.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme32a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( N  .\/  ( X 
./\  W ) ) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z   
y, C    D, s,
y, z    y, E    H, s, t    .\/ , s,
t, x, y, z    K, s, t    .<_ , s, t, x, y, z    ./\ , s,
t, x, y, z   
x, N, z    P, s, t, x, y, z    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    X, s, t, x, z   
y, H    y, K    z, H    z, K
Allowed substitution hints:    C( x, z, t, s)    D( x, t)    E( x, z, t, s)    F( x, y, 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 cdleme32a
StepHypRef Expression
1 cdleme32.b . . . 4  |-  B  =  ( Base `  K
)
2 fvex 5858 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2486 . . 3  |-  B  e. 
_V
4 anass 647 . . . 4  |-  ( ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
5 cdleme32.o . . . . . . 7  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
6 cdleme32.f . . . . . . 7  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
7 eqid 2402 . . . . . . 7  |-  ( 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 ) ) ) )
85, 6, 7cdleme31fv1 33390 . . . . . 6  |-  ( ( 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 ) ) ) ) )
98adantl 464 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( 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 ) ) ) ) )
10 cdleme32.l . . . . . . 7  |-  .<_  =  ( le `  K )
11 cdleme32.j . . . . . . 7  |-  .\/  =  ( join `  K )
12 cdleme32.m . . . . . . 7  |-  ./\  =  ( meet `  K )
13 cdleme32.a . . . . . . 7  |-  A  =  ( Atoms `  K )
14 cdleme32.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
15 cdleme32.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
16 cdleme32.c . . . . . . 7  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
17 cdleme32.d . . . . . . 7  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
18 cdleme32.e . . . . . . 7  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
19 cdleme32.i . . . . . . 7  |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
20 cdleme32.n . . . . . . 7  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
211, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 5, 6cdleme32fvcl 33439 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( F `  X
)  e.  B )
2221adantrr 715 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )  ->  ( F `  X )  e.  B )
239, 22riotasvd 31960 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )  /\  B  e.  _V )  ->  (
( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( N  .\/  ( X 
./\  W ) ) ) )
244, 23syl5bi 217 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )  /\  B  e.  _V )  ->  (
( ( s  e.  A  /\  -.  s  .<_  W )  /\  (
s  .\/  ( X  ./\ 
W ) )  =  X )  ->  ( F `  X )  =  ( N  .\/  ( X  ./\  W ) ) ) )
253, 24mpan2 669 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )  ->  (
( ( s  e.  A  /\  -.  s  .<_  W )  /\  (
s  .\/  ( X  ./\ 
W ) )  =  X )  ->  ( F `  X )  =  ( N  .\/  ( X  ./\  W ) ) ) )
26253impia 1194 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( N  .\/  ( X 
./\  W ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   _Vcvv 3058   ifcif 3884   class class class wbr 4394    |-> cmpt 4452   ` cfv 5568   iota_crio 6238  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Atomscatm 32261   HLchlt 32348   LHypclh 32981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985
This theorem is referenced by:  cdleme32b  33441  cdleme32c  33442  cdleme32e  33444
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