cdlemk15-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 21 on p. 119. Q , C are k_2, f_2. (Contributed by NM, 1-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 )
Assertion	cdlemk15-2N	\|- ( ( ( 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 ) ) ) -> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( F o. `' C ) ) ) ) )

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		simp11	\|- ( ( ( 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 )
12		simp12	\|- ( ( ( 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 )
13	11 12	jca	\|- ( ( ( 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 ) )
14		simp21	\|- ( ( ( 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 )
15		simp22	\|- ( ( ( 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 )
16		simp23	\|- ( ( ( 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 )
17		simp33	\|- ( ( ( 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 ) )
18		simp13	\|- ( ( ( 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 ) )
19		simp32l	\|- ( ( ( 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 ) )
20		simp32r	\|- ( ( ( 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 ) )
21		simp31	\|- ( ( ( 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 ) )
22	1 2 3 4 5 6 7 8 9 10	cdlemk15	\|- ( ( ( ( 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 ) ) ) -> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( F o. `' C ) ) ) ) )
23	13 14 15 16 17 18 19 20 21 22	syl333anc	\|- ( ( ( 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 ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( F o. `' C ) ) ) ) )

Metamath Proof Explorer

Theorem cdlemk15-2N

Proof