Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme17d2 Unicode version

Theorem cdleme17d2 29814
Description: Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph.  F,  G represent f(s), fs(p) respectively. We show, in their notation, fs(p)=q. TODO FIX COMMENT (Contributed by NM, 5-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46.b  |-  B  =  ( Base `  K
)
cdlemef46.l  |-  .<_  =  ( le `  K )
cdlemef46.j  |-  .\/  =  ( join `  K )
cdlemef46.m  |-  ./\  =  ( meet `  K )
cdlemef46.a  |-  A  =  ( Atoms `  K )
cdlemef46.h  |-  H  =  ( LHyp `  K
)
cdlemef46.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef46.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs46.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef46.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
Assertion
Ref Expression
cdleme17d2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F `  P )  =  Q )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, 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    S, s, t, x, y, z
Allowed substitution hints:    D( t)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme17d2
StepHypRef Expression
1 simp1 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp2l 986 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  Q )
3 simp12 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2r 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
5 simp11l 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
6 simp12l 1073 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
7 simp13l 1075 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
8 cdlemef46.l . . . . 5  |-  .<_  =  ( le `  K )
9 cdlemef46.j . . . . 5  |-  .\/  =  ( join `  K )
10 cdlemef46.a . . . . 5  |-  A  =  ( Atoms `  K )
118, 9, 10hlatlej1 28694 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
125, 6, 7, 11syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  .<_  ( P  .\/  Q
) )
13 simp3 962 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  -.  S  .<_  ( P  .\/  Q ) )
14 cdlemef46.b . . . 4  |-  B  =  ( Base `  K
)
15 cdlemef46.m . . . 4  |-  ./\  =  ( meet `  K )
16 cdlemef46.h . . . 4  |-  H  =  ( LHyp `  K
)
17 cdlemef46.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdlemef46.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
19 cdlemef46.f . . . 4  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
20 cdlemefs46.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
2114, 8, 9, 15, 10, 16, 17, 18, 19, 20cdlemefs45 29748 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( F `  P )  =  [_ P  /  s ]_ [_ S  /  t ]_ E
)
221, 2, 3, 4, 12, 13, 21syl132anc 1205 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F `  P )  =  [_ P  /  s ]_ [_ S  /  t ]_ E )
23 simp2rl 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  e.  A )
24 eqid 2256 . . . 4  |-  ( ( S  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  =  ( ( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
25 eqid 2256 . . . 4  |-  ( ( P  .\/  Q ) 
./\  ( ( ( S  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )
2618, 20, 24, 25cdleme31sde 29704 . . 3  |-  ( ( P  e.  A  /\  S  e.  A )  ->  [_ P  /  s ]_ [_ S  /  t ]_ E  =  (
( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
276, 23, 26syl2anc 645 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  [_ P  /  s ]_ [_ S  /  t ]_ E  =  ( ( P 
.\/  Q )  ./\  ( ( ( S 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )  .\/  ( ( P  .\/  S ) 
./\  W ) ) ) )
28 simp11 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
298, 9, 15, 10, 16, 17, 24, 25cdleme17d1 29608 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )  =  Q )
3028, 3, 7, 4, 13, 29syl131anc 1200 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )  =  Q )
3122, 27, 303eqtrd 2292 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F `  P )  =  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   [_csb 3023   ifcif 3506   class class class wbr 3963    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme17d3  29815
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307
  Copyright terms: Public domain W3C validator