Metamath Proof Explorer

Theorem cdleme40w

Description: Part of proof of Lemma E in Crawley p. 113. Apply cdleme40v bound variable change to [_ S / u ]_ V . TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013)

Ref Expression
Hypotheses cdleme40.b 
|- B = ( Base ` K )
cdleme40.l 
|- .<_ = ( le ` K )
cdleme40.j 
|- .\/ = ( join ` K )
cdleme40.m 
|- ./\ = ( meet ` K )
cdleme40.a 
|- A = ( Atoms ` K )
cdleme40.h 
|- H = ( LHyp ` K )
cdleme40.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme40.e 
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme40.g 
|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme40.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme40.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme40.d 
|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme40r.y 
|- Y = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
Assertion cdleme40w 
|- ( ( ( ( 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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )

Proof

Step Hyp Ref Expression
1 cdleme40.b 
 |-  B = ( Base ` K )
2 cdleme40.l 
 |-  .<_ = ( le ` K )
3 cdleme40.j 
 |-  .\/ = ( join ` K )
4 cdleme40.m 
 |-  ./\ = ( meet ` K )
5 cdleme40.a 
 |-  A = ( Atoms ` K )
6 cdleme40.h 
 |-  H = ( LHyp ` K )
7 cdleme40.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme40.e 
 |-  E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdleme40.g 
 |-  G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdleme40.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
11 cdleme40.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , D )
12 cdleme40.d 
 |-  D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
13 cdleme40r.y 
 |-  Y = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
14 eqid 
 |-  ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) )
15 eqid 
 |-  ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) ) ) ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) ) ) )
16 eqid 
 |-  ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) = ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) )
17 eqid 
 |-  ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( S .\/ v ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( S .\/ v ) ./\ W ) ) )
18 eqid 
 |-  ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) )
19 eqid 
 |-  ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) = ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) )
20 eqid 
 |-  if ( u .<_ ( P .\/ Q ) , ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) ) = if ( u .<_ ( P .\/ Q ) , ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) )
21 eqid 
 |-  ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( S .\/ v ) ./\ W ) ) ) ) ) = ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( S .\/ v ) ./\ W ) ) ) ) )
22 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 19 20 21 cdleme40n 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / u ]_ if ( u .<_ ( P .\/ Q ) , ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) ) )
23 simp23l 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> S e. A )
24 eqid 
 |-  ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
25 1 2 3 4 5 6 7 8 9 10 11 12 24 16 18 19 20 cdleme40v 
 |-  ( S e. A -> [_ S / s ]_ N = [_ S / u ]_ if ( u .<_ ( P .\/ Q ) , ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) ) )
26 23 25 syl 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ S / s ]_ N = [_ S / u ]_ if ( u .<_ ( P .\/ Q ) , ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = ( ( P .\/ Q ) ./\ ( ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) ) )
27 22 26 neeqtrrd 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )