cdlemk19

Description: Part of proof of Lemma K of Crawley p. 118. Line 22 on p. 119. N , U , O , D are k, sigma_1 (p), k_1, f_1. (Contributed by NM, 2-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk1.b	\|- B = ( Base ` K )
	cdlemk1.l	\|- .<_ = ( le ` K )
	cdlemk1.j	\|- .\/ = ( join ` K )
	cdlemk1.m	\|- ./\ = ( meet ` K )
	cdlemk1.a	\|- A = ( Atoms ` K )
	cdlemk1.h	\|- H = ( LHyp ` K )
	cdlemk1.t	\|- T = ( ( LTrn ` K ) ` W )
	cdlemk1.r	\|- R = ( ( trL ` K ) ` W )
	cdlemk1.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
	cdlemk1.o	\|- O = ( S ` D )
	cdlemk1.u	\|- U = ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
Assertion	cdlemk19	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( U ` F ) = N )

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	\|- B = ( Base ` K )
2		cdlemk1.l	\|- .<_ = ( le ` K )
3		cdlemk1.j	\|- .\/ = ( join ` K )
4		cdlemk1.m	\|- ./\ = ( meet ` K )
5		cdlemk1.a	\|- A = ( Atoms ` K )
6		cdlemk1.h	\|- H = ( LHyp ` K )
7		cdlemk1.t	\|- T = ( ( LTrn ` K ) ` W )
8		cdlemk1.r	\|- R = ( ( trL ` K ) ` W )
9		cdlemk1.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		cdlemk1.o	\|- O = ( S ` D )
11		cdlemk1.u	\|- U = ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
12		simp11	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
13		simp21	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> N e. T )
14		simp23	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` F ) = ( R ` N ) )
15		simp12	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> F e. T )
16		simp13	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> D e. T )
17		simp33	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` D ) =/= ( R ` F ) )
18	17 17	jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
19		simp31	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> F =/= ( _I \|` B ) )
20		simp32	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> D =/= ( _I \|` B ) )
21	19 19 20	3jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( F =/= ( _I \|` B ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) )
22		simp22	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P e. A /\ -. P .<_ W ) )
23	1 2 3 4 5 6 7 8 9 10 11	cdlemkuel	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ F e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( F =/= ( _I \|` B ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( U ` F ) e. T )
24	12 14 15 15 16 13 18 21 22 23	syl333anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( U ` F ) e. T )
25	1 2 3 4 5 6 7 8 9 10 11	cdlemk18	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) = ( ( U ` F ) ` P ) )
26	2 5 6 7	cdlemd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ ( U ` F ) e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( N ` P ) = ( ( U ` F ) ` P ) ) -> N = ( U ` F ) )
27	12 13 24 22 25 26	syl311anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> N = ( U ` F ) )
28	27	eqcomd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( U ` F ) = N )

Metamath Proof Explorer

Theorem cdlemk19

Proof