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Metamath Proof Explorer

Theorem tdeglem2

Description: Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015)

Ref Expression
Assertion tdeglem2 
|- ( h e. ( NN0 ^m 1o ) |-> ( h ` (/) ) ) = ( h e. ( NN0 ^m 1o ) |-> ( CCfld gsum h ) )

Proof

Step Hyp Ref Expression
1 elmapi 
 |-  ( h e. ( NN0 ^m { (/) } ) -> h : { (/) } --> NN0 )
2 1 feqmptd 
 |-  ( h e. ( NN0 ^m { (/) } ) -> h = ( x e. { (/) } |-> ( h ` x ) ) )
3 2 oveq2d 
 |-  ( h e. ( NN0 ^m { (/) } ) -> ( CCfld gsum h ) = ( CCfld gsum ( x e. { (/) } |-> ( h ` x ) ) ) )
4 cnring 
 |-  CCfld e. Ring
5 ringmnd 
 |-  ( CCfld e. Ring -> CCfld e. Mnd )
6 4 5 mp1i 
 |-  ( h e. ( NN0 ^m { (/) } ) -> CCfld e. Mnd )
7 0ex 
 |-  (/) e. _V
8 7 a1i 
 |-  ( h e. ( NN0 ^m { (/) } ) -> (/) e. _V )
9 7 snid 
 |-  (/) e. { (/) }
10 ffvelcdm 
 |-  ( ( h : { (/) } --> NN0 /\ (/) e. { (/) } ) -> ( h ` (/) ) e. NN0 )
11 1 9 10 sylancl 
 |-  ( h e. ( NN0 ^m { (/) } ) -> ( h ` (/) ) e. NN0 )
12 11 nn0cnd 
 |-  ( h e. ( NN0 ^m { (/) } ) -> ( h ` (/) ) e. CC )
13 cnfldbas 
 |-  CC = ( Base ` CCfld )
14 fveq2 
 |-  ( x = (/) -> ( h ` x ) = ( h ` (/) ) )
15 13 14 gsumsn 
 |-  ( ( CCfld e. Mnd /\ (/) e. _V /\ ( h ` (/) ) e. CC ) -> ( CCfld gsum ( x e. { (/) } |-> ( h ` x ) ) ) = ( h ` (/) ) )
16 6 8 12 15 syl3anc 
 |-  ( h e. ( NN0 ^m { (/) } ) -> ( CCfld gsum ( x e. { (/) } |-> ( h ` x ) ) ) = ( h ` (/) ) )
17 3 16 eqtrd 
 |-  ( h e. ( NN0 ^m { (/) } ) -> ( CCfld gsum h ) = ( h ` (/) ) )
18 df1o2 
 |-  1o = { (/) }
19 18 oveq2i 
 |-  ( NN0 ^m 1o ) = ( NN0 ^m { (/) } )
20 17 19 eleq2s 
 |-  ( h e. ( NN0 ^m 1o ) -> ( CCfld gsum h ) = ( h ` (/) ) )
21 20 eqcomd 
 |-  ( h e. ( NN0 ^m 1o ) -> ( h ` (/) ) = ( CCfld gsum h ) )
22 21 mpteq2ia 
 |-  ( h e. ( NN0 ^m 1o ) |-> ( h ` (/) ) ) = ( h e. ( NN0 ^m 1o ) |-> ( CCfld gsum h ) )