Metamath Proof Explorer


Theorem cdlemk7u-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, p. 119 for the sigma_2 case. (Contributed by NM, 5-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses 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 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 ) )

Proof

Step Hyp Ref Expression
1 cdlemk2.b
 |-  B = ( Base ` K )
2 cdlemk2.l
 |-  .<_ = ( le ` K )
3 cdlemk2.j
 |-  .\/ = ( join ` K )
4 cdlemk2.m
 |-  ./\ = ( meet ` K )
5 cdlemk2.a
 |-  A = ( Atoms ` K )
6 cdlemk2.h
 |-  H = ( LHyp ` K )
7 cdlemk2.t
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk2.r
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk2.s
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk2.q
 |-  Q = ( S ` C )
11 cdlemk2.v
 |-  V = ( d e. T |-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
12 cdlemk2.z
 |-  Z = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' C ) ) .\/ ( R ` ( X o. `' C ) ) ) )
13 simp11
 |-  ( ( ( 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 )
14 simp12
 |-  ( ( ( 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 )
15 13 14 jca
 |-  ( ( ( 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 ) )
16 simp211
 |-  ( ( ( 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 )
17 simp212
 |-  ( ( ( 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 )
18 simp213
 |-  ( ( ( 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 )
19 simp22l
 |-  ( ( ( 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 )
20 simp23l
 |-  ( ( ( 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 )
21 18 19 20 3jca
 |-  ( ( ( 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 ) )
22 simp33
 |-  ( ( ( 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 ) )
23 simp13
 |-  ( ( ( 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 ) )
24 simp32l
 |-  ( ( ( 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 ) )
25 simp32r
 |-  ( ( ( 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 ) )
26 simp22r
 |-  ( ( ( 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 ) )
27 24 25 26 3jca
 |-  ( ( ( 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 ) ) )
28 simp23r
 |-  ( ( ( 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 ) )
29 simp31
 |-  ( ( ( 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 ) ) )
30 1 2 3 4 5 6 7 8 9 10 11 12 cdlemk7u
 |-  ( ( ( ( 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 ) )
31 15 16 17 21 22 23 27 28 29 30 syl333anc
 |-  ( ( ( 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 ) )