Theorem cdleme41fva11

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(r) is one-to-one for r in W (r an atom not under w). TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013)

	Ref	Expression
Hypotheses	cdleme41.b	|- B = ( Base ` K )
	cdleme41.l	|- .<_ = ( le ` K )
	cdleme41.j	|- .\/ = ( join ` K )
	cdleme41.m	|- ./\ = ( meet ` K )
	cdleme41.a	|- A = ( Atoms ` K )
	cdleme41.h	|- H = ( LHyp ` K )
	cdleme41.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdleme41.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
	cdleme41.e	|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdleme41.g	|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
	cdleme41.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
	cdleme41.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
	cdleme41.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
	cdleme41.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
Assertion	cdleme41fva11	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( F ` R ) =/= ( F ` S ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme41.b	|- B = ( Base ` K )
2		cdleme41.l	|- .<_ = ( le ` K )
3		cdleme41.j	|- .\/ = ( join ` K )
4		cdleme41.m	|- ./\ = ( meet ` K )
5		cdleme41.a	|- A = ( Atoms ` K )
6		cdleme41.h	|- H = ( LHyp ` K )
7		cdleme41.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdleme41.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9		cdleme41.e	|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10		cdleme41.g	|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
11		cdleme41.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
12		cdleme41.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
13		cdleme41.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
14		cdleme41.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
15	1 2 3 4 5 6 7 8 9 10 11 12	cdleme41snaw	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
16		simp1	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
17		simp22	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( R e. A /\ -. R .<_ W ) )
18		simp21	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> P =/= Q )
19	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32fva1	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> ( F ` R ) = [_ R / s ]_ N )
20	16 17 18 19	syl3anc	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( F ` R ) = [_ R / s ]_ N )
21		simp23	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( S e. A /\ -. S .<_ W ) )
22	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32fva1	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ -. S .<_ W ) /\ P =/= Q ) -> ( F ` S ) = [_ S / s ]_ N )
23	16 21 18 22	syl3anc	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( F ` S ) = [_ S / s ]_ N )
24	15 20 23	3netr4d	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( F ` R ) =/= ( F ` S ) )