cdlemk19u

Description: Part of Lemma K of Crawley p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with F , N , U . (Contributed by NM, 31-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk5.b	\|- B = ( Base ` K )
	cdlemk5.l	\|- .<_ = ( le ` K )
	cdlemk5.j	\|- .\/ = ( join ` K )
	cdlemk5.m	\|- ./\ = ( meet ` K )
	cdlemk5.a	\|- A = ( Atoms ` K )
	cdlemk5.h	\|- H = ( LHyp ` K )
	cdlemk5.t	\|- T = ( ( LTrn ` K ) ` W )
	cdlemk5.r	\|- R = ( ( trL ` K ) ` W )
	cdlemk5.z	\|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
	cdlemk5.y	\|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
	cdlemk5.x	\|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
	cdlemk5.u	\|- U = ( g e. T \|-> if ( F = N , g , X ) )
Assertion	cdlemk19u	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( U ` F ) = N )

Proof

Step	Hyp	Ref	Expression
1		cdlemk5.b	\|- B = ( Base ` K )
2		cdlemk5.l	\|- .<_ = ( le ` K )
3		cdlemk5.j	\|- .\/ = ( join ` K )
4		cdlemk5.m	\|- ./\ = ( meet ` K )
5		cdlemk5.a	\|- A = ( Atoms ` K )
6		cdlemk5.h	\|- H = ( LHyp ` K )
7		cdlemk5.t	\|- T = ( ( LTrn ` K ) ` W )
8		cdlemk5.r	\|- R = ( ( trL ` K ) ` W )
9		cdlemk5.z	\|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10		cdlemk5.y	\|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11		cdlemk5.x	\|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12		cdlemk5.u	\|- U = ( g e. T \|-> if ( F = N , g , X ) )
13		simp1l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
14		simp1	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
15		simp2l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
16		simp2r	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> N e. T )
17		simp3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
18	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk35u	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( U ` F ) e. T )
19	14 15 16 15 17 18	syl131anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( U ` F ) e. T )
20		simpr	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> F = N )
21		simpl2l	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> F e. T )
22	11 12	cdlemk40t	\|- ( ( F = N /\ F e. T ) -> ( U ` F ) = F )
23	20 21 22	syl2anc	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( U ` F ) = F )
24	23	fveq1d	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( ( U ` F ) ` P ) = ( F ` P ) )
25		fveq1	\|- ( F = N -> ( F ` P ) = ( N ` P ) )
26	25	adantl	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( F ` P ) = ( N ` P ) )
27	24 26	eqtrd	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( ( U ` F ) ` P ) = ( N ` P ) )
28		simpl1	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
29		simpl2l	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> F e. T )
30		simpr	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> F =/= N )
31		simpl2r	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> N e. T )
32		simpl3	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( P e. A /\ -. P .<_ W ) )
33	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk19u1	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( N ` P ) )
34	28 29 30 31 32 33	syl131anc	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( ( U ` F ) ` P ) = ( N ` P ) )
35	27 34	pm2.61dane	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( N ` P ) )
36	2 5 6 7	cdlemd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( U ` F ) e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( U ` F ) ` P ) = ( N ` P ) ) -> ( U ` F ) = N )
37	13 19 16 17 35 36	syl311anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( U ` F ) = N )

Metamath Proof Explorer

Theorem cdlemk19u

Proof