Metamath Proof Explorer

Theorem cdleme50rn

Description: Part of proof of Lemma D in Crawley p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013)

		Ref	Expression
	Hypotheses	cdlemef50.b	\|- B = ( Base ` K )
		cdlemef50.l	\|- .<_ = ( le ` K )
		cdlemef50.j	\|- .\/ = ( join ` K )
		cdlemef50.m	\|- ./\ = ( meet ` K )
		cdlemef50.a	\|- A = ( Atoms ` K )
		cdlemef50.h	\|- H = ( LHyp ` K )
		cdlemef50.u	\|- U = ( ( P .\/ Q ) ./\ W )
		cdlemef50.d	\|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
		cdlemefs50.e	\|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
		cdlemef50.f	\|- F = ( x e. B \|-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
	Assertion	cdleme50rn	\|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ran F = B )

Proof

Step	Hyp	Ref	Expression
1		cdlemef50.b	\|- B = ( Base ` K )
2		cdlemef50.l	\|- .<_ = ( le ` K )
3		cdlemef50.j	\|- .\/ = ( join ` K )
4		cdlemef50.m	\|- ./\ = ( meet ` K )
5		cdlemef50.a	\|- A = ( Atoms ` K )
6		cdlemef50.h	\|- H = ( LHyp ` K )
7		cdlemef50.u	\|- U = ( ( P .\/ Q ) ./\ W )
8		cdlemef50.d	\|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemefs50.e	\|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10		cdlemef50.f	\|- F = ( x e. B \|-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11		eqid	\|- ( ( Q .\/ P ) ./\ W ) = ( ( Q .\/ P ) ./\ W )
12		eqid	\|- ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) = ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
13		eqid	\|- ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) )
14		eqid	\|- ( a e. B \|-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , [_ u / v ]_ ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) ) .\/ ( a ./\ W ) ) ) ) , a ) ) = ( a e. B \|-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , [_ u / v ]_ ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) ) .\/ ( a ./\ W ) ) ) ) , a ) )
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme50rnlem	\|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ran F = B )