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Theorem cdlemk16 34820
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-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
)
Assertion
Ref Expression
cdlemk16  |-  ( ( ( ( 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  .\/  ( R `
 F ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( F  o.  `' D ) ) ) )  e.  A  /\  -.  (
( P  .\/  ( R `  F )
)  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) ) 
.<_  W ) )
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
Allowed substitution hints:    A( f)    B( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)    O( f, i)

Proof of Theorem cdlemk16
StepHypRef Expression
1 simp11 1018 . 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 ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp23 1023 . 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 ) ) )  ->  ( R `  F )  =  ( R `  N ) )
3 simp12 1019 . 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 ) ) )  ->  F  e.  T
)
4 simp13 1020 . 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 ) ) )  ->  D  e.  T
)
5 simp21 1021 . 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  e.  T
)
6 simp33 1026 . . 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 )
)
76, 6jca 532 . 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 ) ) )  ->  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  F )
) )
8 simp31 1024 . . 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 ) )
9 simp32 1025 . . 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  =/=  (  _I  |`  B ) )
108, 8, 93jca 1168 . 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 ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
11 simp22 1022 . 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 ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 cdlemk1.b . . 3  |-  B  =  ( Base `  K
)
13 cdlemk1.l . . 3  |-  .<_  =  ( le `  K )
14 cdlemk1.j . . 3  |-  .\/  =  ( join `  K )
15 cdlemk1.m . . 3  |-  ./\  =  ( meet `  K )
16 cdlemk1.a . . 3  |-  A  =  ( Atoms `  K )
17 cdlemk1.h . . 3  |-  H  =  ( LHyp `  K
)
18 cdlemk1.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
19 cdlemk1.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
20 cdlemk1.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
21 cdlemk1.o . . 3  |-  O  =  ( S `  D
)
2212, 13, 14, 15, 16, 17, 18, 19, 20, 21cdlemk16a 34819 . 2  |-  ( ( ( ( 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 ) ) )  ->  ( (
( P  .\/  ( R `  F )
)  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) )  e.  A  /\  -.  ( ( P  .\/  ( R `  F ) )  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) ) 
.<_  W ) )
231, 2, 3, 3, 4, 5, 7, 10, 11, 22syl333anc 1251 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 ) ) )  ->  ( ( ( P  .\/  ( R `
 F ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( F  o.  `' D ) ) ) )  e.  A  /\  -.  (
( P  .\/  ( R `  F )
)  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D
) ) ) ) 
.<_  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395    |-> cmpt 4453    _I cid 4734   `'ccnv 4942    |` cres 4945    o. ccom 4947   ` cfv 5521   iota_crio 6155  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  cdlemk17  34821  cdlemk16-2N  34843
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