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Theorem cdlemk18 30325
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119.  N,  U,  O,  D are k, sigma1 (p), k1, f1. (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
cdlemk18  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P )  =  ( ( U `  F
) `  P )
)
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, 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    e, F
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)

Proof of Theorem cdlemk18
StepHypRef Expression
1 cdlemk1.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemk1.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemk1.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemk1.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemk1.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemk1.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemk1.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk1.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk1.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
10 cdlemk1.o . . 3  |-  O  =  ( S `  D
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cdlemk17 30315 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P )  =  ( ( P  .\/  ( R `  F )
)  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) ) )
12 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp23 992 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( R `  F )  =  ( R `  N ) )
14 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  F  e.  T
)
15 simp13 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  D  e.  T
)
16 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  N  e.  T
)
17 simp33 995 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( R `  D )  =/=  ( R `  F )
)
1817, 17jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  F )
) )
19 simp31 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  F  =/=  (  _I  |`  B ) )
20 simp32 994 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  D  =/=  (  _I  |`  B ) )
2119, 19, 203jca 1134 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
22 simp22 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
23 cdlemk1.u . . . 4  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23cdlemkuv2 30324 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  F  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  F )
)  /\  ( F  =/=  (  _I  |`  B )  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  F ) `  P )  =  ( ( P  .\/  ( R `  F )
)  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) ) )
2512, 13, 14, 14, 15, 16, 18, 21, 22, 24syl333anc 1216 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( U `
 F ) `  P )  =  ( ( P  .\/  ( R `  F )
)  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) ) )
2611, 25eqtr4d 2320 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P )  =  ( ( U `  F
) `  P )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   class class class wbr 4025    e. cmpt 4079    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5222  (class class class)co 5820   iota_crio 6291   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Atomscatm 28721   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   trLctrl 29615
This theorem is referenced by:  cdlemk19  30326  cdlemk18-2N  30343
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616
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