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Theorem cdleme32a 31077
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 5733 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2505 . . 3  |-  B  e. 
_V
4 anass 631 . . . 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 2435 . . . . . . 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 31027 . . . . . 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 453 . . . . 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 31076 . . . . . 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 698 . . . . 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 6583 . . . 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 209 . . 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 653 . 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 1150 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948   ifcif 3731   class class class wbr 4204    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdleme32b  31078  cdleme32c  31079  cdleme32e  31081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624
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