Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk7u-2N Structured version   Unicode version

Theorem cdlemk7u-2N 33887
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma2 case. (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 ) ) ) ) ) )
cdlemk2.z  |-  Z  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' C ) )  .\/  ( R `  ( X  o.  `' C ) ) ) )
Assertion
Ref Expression
cdlemk7u-2N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  G ) `  P )  .<_  ( ( ( V `  X
) `  P )  .\/  Z ) )
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    G, 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    X, d, k
Allowed substitution hints:    A( f, d)    B( f, i, k, d)    Q( f, i)    S( f, i, k, d)    G( f, i)    H( f, d)    K( f, d)    .<_ ( f, d)    N( d)    V( f, i, k, d)    X( f, i)    Z( f, i, k, d)

Proof of Theorem cdlemk7u-2N
StepHypRef Expression
1 simp11 1027 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp12 1028 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
31, 2jca 530 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp211 1135 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T )
5 simp212 1136 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  e.  T )
6 simp213 1137 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T )
7 simp22l 1116 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T )
8 simp23l 1118 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  X  e.  T )
96, 7, 83jca 1177 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N  e.  T  /\  G  e.  T  /\  X  e.  T ) )
10 simp33 1035 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
11 simp13 1029 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
12 simp32l 1122 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
13 simp32r 1123 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  =/=  (  _I  |`  B ) )
14 simp22r 1117 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  =/=  (  _I  |`  B ) )
1512, 13, 143jca 1177 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
16 simp23r 1119 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  X  =/=  (  _I  |`  B ) )
17 simp31 1033 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( R `  C )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  X
)  =/=  ( R `
 C ) ) )
18 cdlemk2.b . . 3  |-  B  =  ( Base `  K
)
19 cdlemk2.l . . 3  |-  .<_  =  ( le `  K )
20 cdlemk2.j . . 3  |-  .\/  =  ( join `  K )
21 cdlemk2.m . . 3  |-  ./\  =  ( meet `  K )
22 cdlemk2.a . . 3  |-  A  =  ( Atoms `  K )
23 cdlemk2.h . . 3  |-  H  =  ( LHyp `  K
)
24 cdlemk2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
25 cdlemk2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
26 cdlemk2.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
27 cdlemk2.q . . 3  |-  Q  =  ( S `  C
)
28 cdlemk2.v . . 3  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
29 cdlemk2.z . . 3  |-  Z  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' C ) )  .\/  ( R `  ( X  o.  `' C ) ) ) )
3018, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29cdlemk7u 33869 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 C )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  X
)  =/=  ( R `
 C ) ) ) )  ->  (
( V `  G
) `  P )  .<_  ( ( ( V `
 X ) `  P )  .\/  Z
) )
313, 4, 5, 9, 10, 11, 15, 16, 17, 30syl333anc 1262 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  e.  T  /\  X  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  X )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  G ) `  P )  .<_  ( ( ( V `  X
) `  P )  .\/  Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394    |-> cmpt 4452    _I cid 4732   `'ccnv 4821    |` cres 4824    o. ccom 4826   ` cfv 5568   iota_crio 6238  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   trLctrl 33156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator