Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemkuv2 Structured version   Visualization version   Unicode version

Theorem cdlemkuv2 34436
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma1 (p) is  U, f1 is  D, and k1 is  O. (Contributed by NM, 2-Jul-2013.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
cdlemk1.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
Assertion
Ref Expression
cdlemkuv2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , e    .\/ , e    D, e, j    e, G, j   
e, O    P, e    R, e    T, e    e, W    ./\ , j    .<_ , j    .\/ , j    A, j    D, j    j, F   
j, H    j, K    j, N    j, O    P, j    R, j    T, j   
j, W
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    F( e)    G( f, i)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)

Proof of Theorem cdlemkuv2
StepHypRef Expression
1 simp13 1041 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T )
2 cdlemk1.b . . . . 5  |-  B  =  ( Base `  K
)
3 cdlemk1.l . . . . 5  |-  .<_  =  ( le `  K )
4 cdlemk1.j . . . . 5  |-  .\/  =  ( join `  K )
5 cdlemk1.a . . . . 5  |-  A  =  ( Atoms `  K )
6 cdlemk1.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 cdlemk1.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk1.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk1.m . . . . 5  |-  ./\  =  ( meet `  K )
10 cdlemk1.u . . . . 5  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
112, 3, 4, 5, 6, 7, 8, 9, 10cdlemksv 34413 . . . 4  |-  ( G  e.  T  ->  ( U `  G )  =  ( iota_ j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
121, 11syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  G )  =  (
iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) ) )
1312eqcomd 2458 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( iota_ j  e.  T  ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  =  ( U `
 G ) )
14 cdlemk1.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
15 cdlemk1.o . . . 4  |-  O  =  ( S `  D
)
162, 3, 4, 9, 5, 6, 7, 8, 14, 15, 10cdlemkuel 34434 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  G )  e.  T
)
17 simp11l 1120 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
18 simp11r 1121 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
19 simp33 1047 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
202, 3, 4, 9, 5, 6, 7, 8, 14, 15cdlemk16a 34425 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  A  /\  -.  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  W ) )
213, 5, 6, 7cdleme 34129 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  A  /\  -.  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  W ) )  ->  E! j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
2217, 18, 19, 20, 21syl211anc 1277 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  E! j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
23 nfcv 2593 . . . . . . 7  |-  F/_ j T
24 nfriota1 6245 . . . . . . 7  |-  F/_ j
( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )
2523, 24nfmpt 4463 . . . . . 6  |-  F/_ j
( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
2610, 25nfcxfr 2591 . . . . 5  |-  F/_ j U
27 nfcv 2593 . . . . 5  |-  F/_ j G
2826, 27nffv 5855 . . . 4  |-  F/_ j
( U `  G
)
29 nfcv 2593 . . . . . 6  |-  F/_ j P
3028, 29nffv 5855 . . . . 5  |-  F/_ j
( ( U `  G ) `  P
)
3130nfeq1 2606 . . . 4  |-  F/ j ( ( U `  G ) `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
32 fveq1 5847 . . . . 5  |-  ( j  =  ( U `  G )  ->  (
j `  P )  =  ( ( U `
 G ) `  P ) )
3332eqeq1d 2454 . . . 4  |-  ( j  =  ( U `  G )  ->  (
( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  <->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
3428, 31, 33riota2f 6259 . . 3  |-  ( ( ( U `  G
)  e.  T  /\  E! j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )  ->  ( ( ( U `  G ) `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  <-> 
( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )  =  ( U `  G ) ) )
3516, 22, 34syl2anc 671 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( U `  G
) `  P )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  <->  ( iota_ j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  =  ( U `
 G ) ) )
3613, 35mpbird 240 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 986    = wceq 1448    e. wcel 1891    =/= wne 2622   E!wreu 2739   class class class wbr 4374    |-> cmpt 4433    _I cid 4722   `'ccnv 4811    |` cres 4814    o. ccom 4816   ` cfv 5561   iota_crio 6237  (class class class)co 6276   Basecbs 15132   lecple 15208   joincjn 16200   meetcmee 16201   Atomscatm 32831   HLchlt 32918   LHypclh 33551   LTrncltrn 33668   trLctrl 33726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-riotaBAD 32527
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-iun 4250  df-iin 4251  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-1st 6781  df-2nd 6782  df-undef 7007  df-map 7461  df-preset 16184  df-poset 16202  df-plt 16215  df-lub 16231  df-glb 16232  df-join 16233  df-meet 16234  df-p0 16296  df-p1 16297  df-lat 16303  df-clat 16365  df-oposet 32744  df-ol 32746  df-oml 32747  df-covers 32834  df-ats 32835  df-atl 32866  df-cvlat 32890  df-hlat 32919  df-llines 33065  df-lplanes 33066  df-lvols 33067  df-lines 33068  df-psubsp 33070  df-pmap 33071  df-padd 33363  df-lhyp 33555  df-laut 33556  df-ldil 33671  df-ltrn 33672  df-trl 33727
This theorem is referenced by:  cdlemk18  34437  cdlemk7u  34439  cdlemk12u  34441  cdlemk21N  34442  cdlemk20  34443  cdlemkuv2-2  34454  cdlemk31  34465  cdlemkuv2-3N  34468
  Copyright terms: Public domain W3C validator