Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme32sn2.d |
|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) |
2 |
|
cdleme31sn2.n |
|- N = if ( s .<_ ( P .\/ Q ) , I , D ) |
3 |
|
cdleme31sn2.c |
|- C = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
4 |
|
eqid |
|- if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) |
5 |
2 4
|
cdleme31sn |
|- ( R e. A -> [_ R / s ]_ N = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) ) |
6 |
5
|
adantr |
|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) ) |
7 |
|
iffalse |
|- ( -. R .<_ ( P .\/ Q ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = [_ R / s ]_ D ) |
8 |
1
|
csbeq2i |
|- [_ R / s ]_ D = [_ R / s ]_ ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) |
9 |
7 8
|
eqtrdi |
|- ( -. R .<_ ( P .\/ Q ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = [_ R / s ]_ ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) ) |
10 |
|
nfcvd |
|- ( R e. A -> F/_ s ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) |
11 |
|
oveq1 |
|- ( s = R -> ( s .\/ U ) = ( R .\/ U ) ) |
12 |
|
oveq2 |
|- ( s = R -> ( P .\/ s ) = ( P .\/ R ) ) |
13 |
12
|
oveq1d |
|- ( s = R -> ( ( P .\/ s ) ./\ W ) = ( ( P .\/ R ) ./\ W ) ) |
14 |
13
|
oveq2d |
|- ( s = R -> ( Q .\/ ( ( P .\/ s ) ./\ W ) ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
15 |
11 14
|
oveq12d |
|- ( s = R -> ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) |
16 |
10 15
|
csbiegf |
|- ( R e. A -> [_ R / s ]_ ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) |
17 |
9 16
|
sylan9eqr |
|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) |
18 |
6 17
|
eqtrd |
|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) |
19 |
18 3
|
eqtr4di |
|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = C ) |