Step |
Hyp |
Ref |
Expression |
1 |
|
tdeglem.a |
|- A = { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin } |
2 |
|
tdeglem.h |
|- H = ( h e. A |-> ( CCfld gsum h ) ) |
3 |
|
cnfldbas |
|- CC = ( Base ` CCfld ) |
4 |
|
cnfld0 |
|- 0 = ( 0g ` CCfld ) |
5 |
|
cnfldadd |
|- + = ( +g ` CCfld ) |
6 |
|
cnring |
|- CCfld e. Ring |
7 |
|
ringcmn |
|- ( CCfld e. Ring -> CCfld e. CMnd ) |
8 |
6 7
|
mp1i |
|- ( ( X e. A /\ Y e. A ) -> CCfld e. CMnd ) |
9 |
|
simpl |
|- ( ( X e. A /\ Y e. A ) -> X e. A ) |
10 |
1
|
psrbagf |
|- ( X e. A -> X : I --> NN0 ) |
11 |
|
nn0sscn |
|- NN0 C_ CC |
12 |
|
fss |
|- ( ( X : I --> NN0 /\ NN0 C_ CC ) -> X : I --> CC ) |
13 |
10 11 12
|
sylancl |
|- ( X e. A -> X : I --> CC ) |
14 |
13
|
adantr |
|- ( ( X e. A /\ Y e. A ) -> X : I --> CC ) |
15 |
14
|
ffnd |
|- ( ( X e. A /\ Y e. A ) -> X Fn I ) |
16 |
9 15
|
fndmexd |
|- ( ( X e. A /\ Y e. A ) -> I e. _V ) |
17 |
1
|
psrbagf |
|- ( Y e. A -> Y : I --> NN0 ) |
18 |
|
fss |
|- ( ( Y : I --> NN0 /\ NN0 C_ CC ) -> Y : I --> CC ) |
19 |
17 11 18
|
sylancl |
|- ( Y e. A -> Y : I --> CC ) |
20 |
19
|
adantl |
|- ( ( X e. A /\ Y e. A ) -> Y : I --> CC ) |
21 |
1
|
psrbagfsupp |
|- ( X e. A -> X finSupp 0 ) |
22 |
21
|
adantr |
|- ( ( X e. A /\ Y e. A ) -> X finSupp 0 ) |
23 |
1
|
psrbagfsupp |
|- ( Y e. A -> Y finSupp 0 ) |
24 |
23
|
adantl |
|- ( ( X e. A /\ Y e. A ) -> Y finSupp 0 ) |
25 |
3 4 5 8 16 14 20 22 24
|
gsumadd |
|- ( ( X e. A /\ Y e. A ) -> ( CCfld gsum ( X oF + Y ) ) = ( ( CCfld gsum X ) + ( CCfld gsum Y ) ) ) |
26 |
1
|
psrbagaddcl |
|- ( ( X e. A /\ Y e. A ) -> ( X oF + Y ) e. A ) |
27 |
|
oveq2 |
|- ( h = ( X oF + Y ) -> ( CCfld gsum h ) = ( CCfld gsum ( X oF + Y ) ) ) |
28 |
|
ovex |
|- ( CCfld gsum ( X oF + Y ) ) e. _V |
29 |
27 2 28
|
fvmpt |
|- ( ( X oF + Y ) e. A -> ( H ` ( X oF + Y ) ) = ( CCfld gsum ( X oF + Y ) ) ) |
30 |
26 29
|
syl |
|- ( ( X e. A /\ Y e. A ) -> ( H ` ( X oF + Y ) ) = ( CCfld gsum ( X oF + Y ) ) ) |
31 |
|
oveq2 |
|- ( h = X -> ( CCfld gsum h ) = ( CCfld gsum X ) ) |
32 |
|
ovex |
|- ( CCfld gsum X ) e. _V |
33 |
31 2 32
|
fvmpt |
|- ( X e. A -> ( H ` X ) = ( CCfld gsum X ) ) |
34 |
|
oveq2 |
|- ( h = Y -> ( CCfld gsum h ) = ( CCfld gsum Y ) ) |
35 |
|
ovex |
|- ( CCfld gsum Y ) e. _V |
36 |
34 2 35
|
fvmpt |
|- ( Y e. A -> ( H ` Y ) = ( CCfld gsum Y ) ) |
37 |
33 36
|
oveqan12d |
|- ( ( X e. A /\ Y e. A ) -> ( ( H ` X ) + ( H ` Y ) ) = ( ( CCfld gsum X ) + ( CCfld gsum Y ) ) ) |
38 |
25 30 37
|
3eqtr4d |
|- ( ( X e. A /\ Y e. A ) -> ( H ` ( X oF + Y ) ) = ( ( H ` X ) + ( H ` Y ) ) ) |