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Theorem cdlemk18-3N 31635
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119.  N,  Y,  O,  D are k, sigma2 (p), k1, f1. (Contributed by NM, 7-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk3.u1  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
Assertion
Ref Expression
cdlemk18-3N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( D Y F ) `  P )  =  ( N `  P ) )
Distinct variable groups:    e, d,
f, i,  ./\    .<_ , i    .\/ , d, e, f, i    A, i    j, d, D, e, f, i    f, F, i    i, H    i, K    f, N, i    P, d, e, f, i    R, d, e, f, i    T, d, e, f, i    W, d, e, f, i    ./\ , j    .<_ , j    .\/ , j    A, j    j, F    j, H    j, K    j, N    P, j    R, j    S, d, e, j    T, j    j, W    F, d, e
Allowed substitution hints:    A( e, f, d)    B( e, f, i, j, d)    S( f, i)    H( e, f, d)    K( e, f, d)    .<_ ( e, f, d)    N( e, d)    Y( e, f, i, j, d)

Proof of Theorem cdlemk18-3N
StepHypRef Expression
1 simp22 991 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  e.  T
)
2 simp21 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T
)
3 cdlemk3.b . . . . 5  |-  B  =  ( Base `  K
)
4 cdlemk3.l . . . . 5  |-  .<_  =  ( le `  K )
5 cdlemk3.j . . . . 5  |-  .\/  =  ( join `  K )
6 cdlemk3.m . . . . 5  |-  ./\  =  ( meet `  K )
7 cdlemk3.a . . . . 5  |-  A  =  ( Atoms `  K )
8 cdlemk3.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 cdlemk3.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdlemk3.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
11 cdlemk3.s . . . . 5  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
12 cdlemk3.u1 . . . . 5  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
13 eqid 2436 . . . . 5  |-  ( S `
 D )  =  ( S `  D
)
14 eqid 2436 . . . . 5  |-  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) )  =  ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
153, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdlemkuu 31630 . . . 4  |-  ( ( D  e.  T  /\  F  e.  T )  ->  ( D Y F )  =  ( ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) ) `  F ) )
161, 2, 15syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( D Y F )  =  ( ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) ) `  F ) )
1716fveq1d 5723 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( D Y F ) `  P )  =  ( ( ( e  e.  T  |->  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) ) `  F
) `  P )
)
183, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14cdlemk18-2N 31621 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N `  P )  =  ( ( ( e  e.  T  |->  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) ) `  F
) `  P )
)
1917, 18eqtr4d 2471 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( D Y F ) `  P )  =  ( N `  P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4205    e. cmpt 4259    _I cid 4486   `'ccnv 4870    |` cres 4873    o. ccom 4875   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076   iota_crio 6535   Basecbs 13462   lecple 13529   joincjn 14394   meetcmee 14395   Atomscatm 29999   HLchlt 30086   LHypclh 30719   LTrncltrn 30836   trLctrl 30893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-map 7013  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-llines 30233  df-lplanes 30234  df-lvols 30235  df-lines 30236  df-psubsp 30238  df-pmap 30239  df-padd 30531  df-lhyp 30723  df-laut 30724  df-ldil 30839  df-ltrn 30840  df-trl 30894
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