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Metamath Proof Explorer

Theorem cdleme48fvg

Description: Remove P =/= Q condition in cdleme48fv . TODO: Can this replace uses of cdleme32a ? TODO: Can this be used to help prove the R or S case where X is an atom? TODO: Can this be proved more directly by eliminating P =/= Q in earlier theorems? Should this replace uses of cdleme48fv ? (Contributed by NM, 23-Apr-2013)

Ref Expression
Hypotheses cdlemef46.b 
|- B = ( Base ` K )
cdlemef46.l 
|- .<_ = ( le ` K )
cdlemef46.j 
|- .\/ = ( join ` K )
cdlemef46.m 
|- ./\ = ( meet ` K )
cdlemef46.a 
|- A = ( Atoms ` K )
cdlemef46.h 
|- H = ( LHyp ` K )
cdlemef46.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef46.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs46.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef46.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 cdleme48fvg 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdlemef46.b 
 |-  B = ( Base ` K )
2 cdlemef46.l 
 |-  .<_ = ( le ` K )
3 cdlemef46.j 
 |-  .\/ = ( join ` K )
4 cdlemef46.m 
 |-  ./\ = ( meet ` K )
5 cdlemef46.a 
 |-  A = ( Atoms ` K )
6 cdlemef46.h 
 |-  H = ( LHyp ` K )
7 cdlemef46.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef46.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs46.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemef46.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 simpl3r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( S .\/ ( X ./\ W ) ) = X )
12 simp3ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> S e. A )
13 12 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> S e. A )
14 1 5 atbase 
 |-  ( S e. A -> S e. B )
15 13 14 syl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> S e. B )
16 10 cdleme31id 
 |-  ( ( S e. B /\ P = Q ) -> ( F ` S ) = S )
17 15 16 sylancom 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( F ` S ) = S )
18 17 oveq1d 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( ( F ` S ) .\/ ( X ./\ W ) ) = ( S .\/ ( X ./\ W ) ) )
19 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> X e. B )
20 10 cdleme31id 
 |-  ( ( X e. B /\ P = Q ) -> ( F ` X ) = X )
21 19 20 sylan 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( F ` X ) = X )
22 11 18 21 3eqtr4rd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) )
23 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
24 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> P =/= Q )
25 simpl2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( X e. B /\ -. X .<_ W ) )
26 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) )
27 1 2 3 4 5 6 7 8 9 10 cdleme48fv 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) )
28 23 24 25 26 27 syl121anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) )
29 22 28 pm2.61dane 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) )