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Theorem cdleme32a 29797
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 5472 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2328 . . 3  |-  B  e. 
_V
4 anass 633 . . . 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 2258 . . . . . . 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 29747 . . . . . 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 454 . . . . 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 29796 . . . . . 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 700 . . . . 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 6315 . . . 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 210 . . 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 655 . 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 1153 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   _Vcvv 2763   ifcif 3539   class class class wbr 3997    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Atomscatm 28620   HLchlt 28707   LHypclh 29340
This theorem is referenced by:  cdleme32b  29798  cdleme32c  29799  cdleme32e  29801
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344
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