cdlemk55

Description: Part of proof of Lemma K of Crawley p. 118. Line 11, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 26-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 ) )
Assertion	cdlemk55	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )

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		simpl1	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
13		simpl21	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) )
14		simpl22	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> G e. T )
15		simpl3	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> ( P e. A /\ -. P .<_ W ) )
16		simpl23	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> I e. T )
17		simpr	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> ( R ` G ) = ( R ` I ) )
18	1 2 3 4 5 6 7 8 9 10 11	cdlemk55b	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
19	12 13 14 15 16 17 18	syl132anc	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) = ( R ` I ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
20		simpl1	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
21		simpl21	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) )
22		simpl22	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> G e. T )
23		simpl3	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> ( P e. A /\ -. P .<_ W ) )
24		simpl23	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> I e. T )
25		simpr	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> ( R ` G ) =/= ( R ` I ) )
26	1 2 3 4 5 6 7 8 9 10 11	cdlemk53	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
27	20 21 22 23 24 25 26	syl132anc	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( R ` G ) =/= ( R ` I ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
28	19 27	pm2.61dane	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )

Metamath Proof Explorer

Theorem cdlemk55

Proof