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Metamath Proof Explorer

Theorem cdleme27b

Description: Lemma for cdleme27N . (Contributed by NM, 3-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 cdleme27b 
|- ( s = t -> C = Y )

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 breq1 
 |-  ( s = t -> ( s .<_ ( P .\/ Q ) <-> t .<_ ( P .\/ Q ) ) )
18 oveq1 
 |-  ( s = t -> ( s .\/ z ) = ( t .\/ z ) )
19 18 oveq1d 
 |-  ( s = t -> ( ( s .\/ z ) ./\ W ) = ( ( t .\/ z ) ./\ W ) )
20 19 oveq2d 
 |-  ( s = t -> ( Z .\/ ( ( s .\/ z ) ./\ W ) ) = ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
21 20 oveq2d 
 |-  ( s = t -> ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) )
22 21 10 14 3eqtr4g 
 |-  ( s = t -> N = O )
23 22 eqeq2d 
 |-  ( s = t -> ( u = N <-> u = O ) )
24 23 imbi2d 
 |-  ( s = t -> ( ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) <-> ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) )
25 24 ralbidv 
 |-  ( s = t -> ( A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) <-> A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) )
26 25 riotabidv 
 |-  ( s = t -> ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) ) = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) )
27 26 11 15 3eqtr4g 
 |-  ( s = t -> D = E )
28 oveq1 
 |-  ( s = t -> ( s .\/ U ) = ( t .\/ U ) )
29 oveq2 
 |-  ( s = t -> ( P .\/ s ) = ( P .\/ t ) )
30 29 oveq1d 
 |-  ( s = t -> ( ( P .\/ s ) ./\ W ) = ( ( P .\/ t ) ./\ W ) )
31 30 oveq2d 
 |-  ( s = t -> ( Q .\/ ( ( P .\/ s ) ./\ W ) ) = ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
32 28 31 oveq12d 
 |-  ( s = t -> ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
33 32 8 13 3eqtr4g 
 |-  ( s = t -> F = G )
34 17 27 33 ifbieq12d 
 |-  ( s = t -> if ( s .<_ ( P .\/ Q ) , D , F ) = if ( t .<_ ( P .\/ Q ) , E , G ) )
35 34 12 16 3eqtr4g 
 |-  ( s = t -> C = Y )