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Metamath Proof Explorer

Theorem cdleme42b

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 6-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 cdleme42b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( [_ R / s ]_ N .\/ ( X ./\ W ) ) )

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 fvexi 
 |-  B e. _V
16 nfv 
 |-  F/ s ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) )
17 nfcsb1v 
 |-  F/_ s [_ R / s ]_ N
18 nfcv 
 |-  F/_ s .\/
19 nfcv 
 |-  F/_ s ( X ./\ W )
20 17 18 19 nfov 
 |-  F/_ s ( [_ R / s ]_ N .\/ ( X ./\ W ) )
21 20 a1i 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> F/_ s ( [_ R / s ]_ N .\/ ( X ./\ W ) ) )
22 nfvd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> F/ s ( -. R .<_ W /\ ( R .\/ ( X ./\ W ) ) = X ) )
23 eqid 
 |-  ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> z = ( N .\/ ( X ./\ W ) ) ) ) = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> z = ( N .\/ ( X ./\ W ) ) ) )
24 13 14 23 cdleme31fv1 
 |-  ( ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) -> ( F ` X ) = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> z = ( N .\/ ( X ./\ W ) ) ) ) )
25 24 3ad2ant2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> z = ( N .\/ ( X ./\ W ) ) ) ) )
26 breq1 
 |-  ( s = R -> ( s .<_ W <-> R .<_ W ) )
27 26 notbid 
 |-  ( s = R -> ( -. s .<_ W <-> -. R .<_ W ) )
28 oveq1 
 |-  ( s = R -> ( s .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) )
29 28 eqeq1d 
 |-  ( s = R -> ( ( s .\/ ( X ./\ W ) ) = X <-> ( R .\/ ( X ./\ W ) ) = X ) )
30 27 29 anbi12d 
 |-  ( s = R -> ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) <-> ( -. R .<_ W /\ ( R .\/ ( X ./\ W ) ) = X ) ) )
31 30 adantl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) /\ s = R ) -> ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) <-> ( -. R .<_ W /\ ( R .\/ ( X ./\ W ) ) = X ) ) )
32 csbeq1a 
 |-  ( s = R -> N = [_ R / s ]_ N )
33 32 oveq1d 
 |-  ( s = R -> ( N .\/ ( X ./\ W ) ) = ( [_ R / s ]_ N .\/ ( X ./\ W ) ) )
34 33 adantl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) /\ s = R ) -> ( N .\/ ( X ./\ W ) ) = ( [_ R / s ]_ N .\/ ( X ./\ W ) ) )
35 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
36 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> X e. B )
37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32fvcl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> ( F ` X ) e. B )
38 35 36 37 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) e. B )
39 simp3ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> R e. A )
40 simp3lr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> -. R .<_ W )
41 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( R .\/ ( X ./\ W ) ) = X )
42 40 41 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( -. R .<_ W /\ ( R .\/ ( X ./\ W ) ) = X ) )
43 16 21 22 25 31 34 38 39 42 riotasv2d 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) /\ B e. _V ) -> ( F ` X ) = ( [_ R / s ]_ N .\/ ( X ./\ W ) ) )
44 15 43 mpan2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( [_ R / s ]_ N .\/ ( X ./\ W ) ) )