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Theorem cdlemkuv2 34404
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 1020 . . . 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 34381 . . . 4  |-  ( G  e.  T  ->  ( U `  G )  =  ( iota_ j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
121, 11syl 16 . . 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 2443 . 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 34402 . . 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 1099 . . . 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 1100 . . . 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 1026 . . . 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 34393 . . . 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 34097 . . . 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 1224 . . 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 2574 . . . . . . 7  |-  F/_ j T
24 nfriota1 6054 . . . . . . 7  |-  F/_ j
( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )
2523, 24nfmpt 4375 . . . . . 6  |-  F/_ j
( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
2610, 25nfcxfr 2571 . . . . 5  |-  F/_ j U
27 nfcv 2574 . . . . 5  |-  F/_ j G
2826, 27nffv 5693 . . . 4  |-  F/_ j
( U `  G
)
29 nfcv 2574 . . . . . 6  |-  F/_ j P
3028, 29nffv 5693 . . . . 5  |-  F/_ j
( ( U `  G ) `  P
)
3130nfeq1 2583 . . . 4  |-  F/ j ( ( U `  G ) `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
32 fveq1 5685 . . . . 5  |-  ( j  =  ( U `  G )  ->  (
j `  P )  =  ( ( U `
 G ) `  P ) )
3332eqeq1d 2446 . . . 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 6069 . . 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 661 . 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 232 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E!wreu 2712   class class class wbr 4287    e. cmpt 4345    _I cid 4626   `'ccnv 4834    |` cres 4837    o. ccom 4839   ` cfv 5413   iota_crio 6046  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   Atomscatm 32801   HLchlt 32888   LHypclh 33521   LTrncltrn 33638   trLctrl 33695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32497
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-undef 6784  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32714  df-ol 32716  df-oml 32717  df-covers 32804  df-ats 32805  df-atl 32836  df-cvlat 32860  df-hlat 32889  df-llines 33035  df-lplanes 33036  df-lvols 33037  df-lines 33038  df-psubsp 33040  df-pmap 33041  df-padd 33333  df-lhyp 33525  df-laut 33526  df-ldil 33641  df-ltrn 33642  df-trl 33696
This theorem is referenced by:  cdlemk18  34405  cdlemk7u  34407  cdlemk12u  34409  cdlemk21N  34410  cdlemk20  34411  cdlemkuv2-2  34422  cdlemk31  34433  cdlemkuv2-3N  34436
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