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 ffvelrn
 |-  ( ( 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 ) )