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Theorem cdlemk16-2N 30199
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk2.b  |-  B  =  ( Base `  K
)
cdlemk2.l  |-  .<_  =  ( le `  K )
cdlemk2.j  |-  .\/  =  ( join `  K )
cdlemk2.m  |-  ./\  =  ( meet `  K )
cdlemk2.a  |-  A  =  ( Atoms `  K )
cdlemk2.h  |-  H  =  ( LHyp `  K
)
cdlemk2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk2.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk2.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk2.q  |-  Q  =  ( S `  C
)
Assertion
Ref Expression
cdlemk16-2N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( ( P  .\/  ( R `
 F ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( F  o.  `' C ) ) ) )  e.  A  /\  -.  (
( P  .\/  ( R `  F )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( F  o.  `' C
) ) ) ) 
.<_  W ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    C, 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)    Q( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)

Proof of Theorem cdlemk16-2N
StepHypRef Expression
1 simp11 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp12 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H
)
31, 2jca 520 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp21 993 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T
)
5 simp22 994 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  e.  T
)
6 simp23 995 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T
)
7 simp33 998 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp13 992 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
9 simp32l 1085 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
10 simp32r 1086 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  =/=  (  _I  |`  B ) )
11 simp31 996 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  C )  =/=  ( R `  F )
)
12 cdlemk2.b . . 3  |-  B  =  ( Base `  K
)
13 cdlemk2.l . . 3  |-  .<_  =  ( le `  K )
14 cdlemk2.j . . 3  |-  .\/  =  ( join `  K )
15 cdlemk2.m . . 3  |-  ./\  =  ( meet `  K )
16 cdlemk2.a . . 3  |-  A  =  ( Atoms `  K )
17 cdlemk2.h . . 3  |-  H  =  ( LHyp `  K
)
18 cdlemk2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
19 cdlemk2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
20 cdlemk2.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
21 cdlemk2.q . . 3  |-  Q  =  ( S `  C
)
2212, 13, 14, 15, 16, 17, 18, 19, 20, 21cdlemk16 30176 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  ( R `  C )  =/=  ( R `  F ) ) )  ->  ( ( ( P  .\/  ( R `
 F ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( F  o.  `' C ) ) ) )  e.  A  /\  -.  (
( P  .\/  ( R `  F )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( F  o.  `' C
) ) ) ) 
.<_  W ) )
233, 4, 5, 6, 7, 8, 9, 10, 11, 22syl333anc 1219 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( ( P  .\/  ( R `
 F ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( F  o.  `' C ) ) ) )  e.  A  /\  -.  (
( P  .\/  ( R `  F )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( F  o.  `' C
) ) ) ) 
.<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963    e. cmpt 4017    _I cid 4241   `'ccnv 4625    |` cres 4628    o. ccom 4630   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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