Step |
Hyp |
Ref |
Expression |
1 |
|
selvval.p |
|- P = ( I mPoly R ) |
2 |
|
selvval.b |
|- B = ( Base ` P ) |
3 |
|
selvval.u |
|- U = ( ( I \ J ) mPoly R ) |
4 |
|
selvval.t |
|- T = ( J mPoly U ) |
5 |
|
selvval.c |
|- C = ( algSc ` T ) |
6 |
|
selvval.d |
|- D = ( C o. ( algSc ` U ) ) |
7 |
|
selvval.j |
|- ( ph -> J C_ I ) |
8 |
|
selvval.f |
|- ( ph -> F e. B ) |
9 |
|
coeq2 |
|- ( f = F -> ( d o. f ) = ( d o. F ) ) |
10 |
9
|
fveq2d |
|- ( f = F -> ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) = ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ) |
11 |
10
|
fveq1d |
|- ( f = F -> ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
12 |
11
|
csbeq2dv |
|- ( f = F -> [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
13 |
12
|
csbeq2dv |
|- ( f = F -> [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
14 |
13
|
csbeq2dv |
|- ( f = F -> [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
15 |
14
|
csbeq2dv |
|- ( f = F -> [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
16 |
|
reldmmpl |
|- Rel dom mPoly |
17 |
16 1 2
|
elbasov |
|- ( F e. B -> ( I e. _V /\ R e. _V ) ) |
18 |
8 17
|
syl |
|- ( ph -> ( I e. _V /\ R e. _V ) ) |
19 |
18
|
simpld |
|- ( ph -> I e. _V ) |
20 |
18
|
simprd |
|- ( ph -> R e. _V ) |
21 |
19 20 7
|
selvfval |
|- ( ph -> ( ( I selectVars R ) ` J ) = ( f e. ( Base ` ( I mPoly R ) ) |-> [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. f ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) ) |
22 |
1
|
fveq2i |
|- ( Base ` P ) = ( Base ` ( I mPoly R ) ) |
23 |
2 22
|
eqtri |
|- B = ( Base ` ( I mPoly R ) ) |
24 |
8 23
|
eleqtrdi |
|- ( ph -> F e. ( Base ` ( I mPoly R ) ) ) |
25 |
|
fvex |
|- ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) e. _V |
26 |
25
|
csbex |
|- [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) e. _V |
27 |
26
|
csbex |
|- [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) e. _V |
28 |
27
|
csbex |
|- [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) e. _V |
29 |
28
|
csbex |
|- [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) e. _V |
30 |
29
|
a1i |
|- ( ph -> [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) e. _V ) |
31 |
15 21 24 30
|
fvmptd4 |
|- ( ph -> ( ( ( I selectVars R ) ` J ) ` F ) = [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
32 |
|
ovex |
|- ( ( I \ J ) mPoly R ) e. _V |
33 |
3
|
eqeq2i |
|- ( u = U <-> u = ( ( I \ J ) mPoly R ) ) |
34 |
|
oveq2 |
|- ( u = U -> ( J mPoly u ) = ( J mPoly U ) ) |
35 |
|
fveq2 |
|- ( u = U -> ( algSc ` u ) = ( algSc ` U ) ) |
36 |
35
|
coeq2d |
|- ( u = U -> ( c o. ( algSc ` u ) ) = ( c o. ( algSc ` U ) ) ) |
37 |
|
oveq2 |
|- ( u = U -> ( J mVar u ) = ( J mVar U ) ) |
38 |
37
|
fveq1d |
|- ( u = U -> ( ( J mVar u ) ` x ) = ( ( J mVar U ) ` x ) ) |
39 |
38
|
ifeq1d |
|- ( u = U -> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) = if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) |
40 |
39
|
mpteq2dv |
|- ( u = U -> ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) = ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) |
41 |
40
|
fveq2d |
|- ( u = U -> ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
42 |
36 41
|
csbeq12dv |
|- ( u = U -> [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
43 |
42
|
csbeq2dv |
|- ( u = U -> [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
44 |
34 43
|
csbeq12dv |
|- ( u = U -> [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( J mPoly U ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
45 |
|
ovex |
|- ( J mPoly U ) e. _V |
46 |
4
|
eqeq2i |
|- ( t = T <-> t = ( J mPoly U ) ) |
47 |
|
fveq2 |
|- ( t = T -> ( algSc ` t ) = ( algSc ` T ) ) |
48 |
|
oveq2 |
|- ( t = T -> ( I evalSub t ) = ( I evalSub T ) ) |
49 |
48
|
fveq1d |
|- ( t = T -> ( ( I evalSub t ) ` ran d ) = ( ( I evalSub T ) ` ran d ) ) |
50 |
49
|
fveq1d |
|- ( t = T -> ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) = ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ) |
51 |
50
|
fveq1d |
|- ( t = T -> ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
52 |
51
|
csbeq2dv |
|- ( t = T -> [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
53 |
47 52
|
csbeq12dv |
|- ( t = T -> [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( algSc ` T ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
54 |
|
fvex |
|- ( algSc ` T ) e. _V |
55 |
5
|
eqeq2i |
|- ( c = C <-> c = ( algSc ` T ) ) |
56 |
|
coeq1 |
|- ( c = C -> ( c o. ( algSc ` U ) ) = ( C o. ( algSc ` U ) ) ) |
57 |
|
fveq1 |
|- ( c = C -> ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) = ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) |
58 |
57
|
ifeq2d |
|- ( c = C -> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) = if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) |
59 |
58
|
mpteq2dv |
|- ( c = C -> ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) = ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) |
60 |
59
|
fveq2d |
|- ( c = C -> ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
61 |
56 60
|
csbeq12dv |
|- ( c = C -> [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = [_ ( C o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
62 |
5
|
fvexi |
|- C e. _V |
63 |
|
fvex |
|- ( algSc ` U ) e. _V |
64 |
62 63
|
coex |
|- ( C o. ( algSc ` U ) ) e. _V |
65 |
6
|
eqeq2i |
|- ( d = D <-> d = ( C o. ( algSc ` U ) ) ) |
66 |
|
rneq |
|- ( d = D -> ran d = ran D ) |
67 |
66
|
fveq2d |
|- ( d = D -> ( ( I evalSub T ) ` ran d ) = ( ( I evalSub T ) ` ran D ) ) |
68 |
|
coeq1 |
|- ( d = D -> ( d o. F ) = ( D o. F ) ) |
69 |
67 68
|
fveq12d |
|- ( d = D -> ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) = ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ) |
70 |
69
|
fveq1d |
|- ( d = D -> ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
71 |
65 70
|
sylbir |
|- ( d = ( C o. ( algSc ` U ) ) -> ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
72 |
64 71
|
csbie |
|- [_ ( C o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) |
73 |
61 72
|
eqtrdi |
|- ( c = C -> [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
74 |
55 73
|
sylbir |
|- ( c = ( algSc ` T ) -> [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
75 |
54 74
|
csbie |
|- [_ ( algSc ` T ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub T ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) |
76 |
53 75
|
eqtrdi |
|- ( t = T -> [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
77 |
46 76
|
sylbir |
|- ( t = ( J mPoly U ) -> [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
78 |
45 77
|
csbie |
|- [_ ( J mPoly U ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` U ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) |
79 |
44 78
|
eqtrdi |
|- ( u = U -> [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
80 |
33 79
|
sylbir |
|- ( u = ( ( I \ J ) mPoly R ) -> [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |
81 |
32 80
|
csbie |
|- [_ ( ( I \ J ) mPoly R ) / u ]_ [_ ( J mPoly u ) / t ]_ [_ ( algSc ` t ) / c ]_ [_ ( c o. ( algSc ` u ) ) / d ]_ ( ( ( ( I evalSub t ) ` ran d ) ` ( d o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar u ) ` x ) , ( c ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) |
82 |
31 81
|
eqtrdi |
|- ( ph -> ( ( ( I selectVars R ) ` J ) ` F ) = ( ( ( ( I evalSub T ) ` ran D ) ` ( D o. F ) ) ` ( x e. I |-> if ( x e. J , ( ( J mVar U ) ` x ) , ( C ` ( ( ( I \ J ) mVar R ) ` x ) ) ) ) ) ) |