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Metamath Proof Explorer

Theorem cdleme28c

Description: Part of proof of Lemma E in Crawley p. 113. Eliminate the s =/= t antecedent in cdleme28b . TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013)

Ref Expression
Hypotheses cdleme26.b 
|- B = ( Base ` K )
cdleme26.l 
|- .<_ = ( le ` K )
cdleme26.j 
|- .\/ = ( join ` K )
cdleme26.m 
|- ./\ = ( meet ` K )
cdleme26.a 
|- A = ( Atoms ` K )
cdleme26.h 
|- H = ( LHyp ` K )
cdleme27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme27.f 
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme27.z 
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme27.n 
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
cdleme27.d 
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
cdleme27.c 
|- C = if ( s .<_ ( P .\/ Q ) , D , F )
cdleme27.g 
|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme27.o 
|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
cdleme27.e 
|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
cdleme27.y 
|- Y = if ( t .<_ ( P .\/ Q ) , E , G )
Assertion cdleme28c 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b 
 |-  B = ( Base ` K )
2 cdleme26.l 
 |-  .<_ = ( le ` K )
3 cdleme26.j 
 |-  .\/ = ( join ` K )
4 cdleme26.m 
 |-  ./\ = ( meet ` K )
5 cdleme26.a 
 |-  A = ( Atoms ` K )
6 cdleme26.h 
 |-  H = ( LHyp ` K )
7 cdleme27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme27.f 
 |-  F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme27.z 
 |-  Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
10 cdleme27.n 
 |-  N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
11 cdleme27.d 
 |-  D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
12 cdleme27.c 
 |-  C = if ( s .<_ ( P .\/ Q ) , D , F )
13 cdleme27.g 
 |-  G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
14 cdleme27.o 
 |-  O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
15 cdleme27.e 
 |-  E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
16 cdleme27.y 
 |-  Y = if ( t .<_ ( P .\/ Q ) , E , G )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme27b 
 |-  ( s = t -> C = Y )
18 17 oveq1d 
 |-  ( s = t -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
19 18 adantl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s = t ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
20 simpl11 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( K e. HL /\ W e. H ) )
21 simpl12 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( P e. A /\ -. P .<_ W ) )
22 simpl13 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( Q e. A /\ -. Q .<_ W ) )
23 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> P =/= Q )
24 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( s e. A /\ -. s .<_ W ) )
25 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( t e. A /\ -. t .<_ W ) )
26 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> s =/= t )
27 simpl31 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( s .\/ ( X ./\ W ) ) = X )
28 simpl32 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( t .\/ ( X ./\ W ) ) = X )
29 27 28 jca 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) )
30 simpl33 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( X e. B /\ -. X .<_ W ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme28b 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
32 20 21 22 23 24 25 26 29 30 31 syl333anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
33 19 32 pm2.61dane 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )