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Theorem cdlemk7u-2N 31370
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 987 . . 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 988 . . 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 519 . 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 1095 . 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 1096 . 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 1097 . . 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 1076 . . 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 1078 . . 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 1134 . 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 995 . 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 989 . 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 1082 . . 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 1083 . . 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 1077 . . 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 1134 . 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 1079 . 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 993 . 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 31352 . 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 1216 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172    e. cmpt 4226    _I cid 4453   `'ccnv 4836    |` cres 4839    o. ccom 4841   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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