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Theorem cdlemk20-2N 34899
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our  D,  C,  O,  Q,  U,  V represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-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
)
cdlemk2.v  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
cdlemk2a.o  |-  O  =  ( S `  D
)
Assertion
Ref Expression
cdlemk20-2N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  D ) `  P )  =  ( O `  P ) )
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    ./\ , d    .\/ , d    C, d, k    Q, d    P, d    R, d    T, d    W, d    ./\ , k    .<_ , k    .\/ , k    A, k    C, k   
k, F    k, H    k, K    k, N    Q, k    P, k    R, k    T, k    k, W    F, d    i, k, f, D, d
Allowed substitution hints:    A( f, d)    B( f, i, k, d)    Q( f, i)    S( f, i, k, d)    H( f, d)    K( f, d)    .<_ ( f, d)    N( d)    O( f, i, k, d)    V( f, i, k, d)

Proof of Theorem cdlemk20-2N
StepHypRef Expression
1 simp11 1018 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp12 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
31, 2jca 532 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp211 1126 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T )
5 simp212 1127 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  e.  T )
6 simp213 1128 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T )
7 simp22l 1107 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  e.  T )
86, 7jca 532 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N  e.  T  /\  D  e.  T ) )
9 simp33 1026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
10 simp13 1020 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp32l 1113 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
12 simp32r 1114 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  =/=  (  _I  |`  B ) )
13 simp22r 1108 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  =/=  (  _I  |`  B ) )
1411, 12, 133jca 1168 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
15 simp31 1024 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( R `  C )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 C ) ) )
16 cdlemk2.b . . 3  |-  B  =  ( Base `  K
)
17 cdlemk2.l . . 3  |-  .<_  =  ( le `  K )
18 cdlemk2.j . . 3  |-  .\/  =  ( join `  K )
19 cdlemk2.m . . 3  |-  ./\  =  ( meet `  K )
20 cdlemk2.a . . 3  |-  A  =  ( Atoms `  K )
21 cdlemk2.h . . 3  |-  H  =  ( LHyp `  K
)
22 cdlemk2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
23 cdlemk2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
24 cdlemk2.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
25 cdlemk2.q . . 3  |-  Q  =  ( S `  C
)
26 cdlemk2.v . . 3  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
27 cdlemk2a.o . . 3  |-  O  =  ( S `  D
)
2816, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27cdlemk20 34881 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  (
( N  e.  T  /\  D  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  C )
) ) )  -> 
( ( V `  D ) `  P
)  =  ( O `
 P ) )
293, 4, 5, 8, 9, 10, 14, 15, 28syl332anc 1250 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  D ) `  P )  =  ( O `  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403    |-> cmpt 4461    _I cid 4742   `'ccnv 4950    |` cres 4953    o. ccom 4955   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Atomscatm 33271   HLchlt 33358   LHypclh 33991   LTrncltrn 34108   trLctrl 34165
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32967
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-map 7329  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166
This theorem is referenced by: (None)
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