Step |
Hyp |
Ref |
Expression |
1 |
|
prodsnf.1 |
|- F/_ k B |
2 |
|
prodsnf.2 |
|- ( k = M -> A = B ) |
3 |
|
nfcv |
|- F/_ m A |
4 |
|
nfcsb1v |
|- F/_ k [_ m / k ]_ A |
5 |
|
csbeq1a |
|- ( k = m -> A = [_ m / k ]_ A ) |
6 |
3 4 5
|
cbvprodi |
|- prod_ k e. { M } A = prod_ m e. { M } [_ m / k ]_ A |
7 |
|
csbeq1 |
|- ( m = ( { <. 1 , M >. } ` n ) -> [_ m / k ]_ A = [_ ( { <. 1 , M >. } ` n ) / k ]_ A ) |
8 |
|
1nn |
|- 1 e. NN |
9 |
8
|
a1i |
|- ( ( M e. V /\ B e. CC ) -> 1 e. NN ) |
10 |
|
1z |
|- 1 e. ZZ |
11 |
|
f1osng |
|- ( ( 1 e. ZZ /\ M e. V ) -> { <. 1 , M >. } : { 1 } -1-1-onto-> { M } ) |
12 |
|
fzsn |
|- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } ) |
13 |
10 12
|
ax-mp |
|- ( 1 ... 1 ) = { 1 } |
14 |
|
f1oeq2 |
|- ( ( 1 ... 1 ) = { 1 } -> ( { <. 1 , M >. } : ( 1 ... 1 ) -1-1-onto-> { M } <-> { <. 1 , M >. } : { 1 } -1-1-onto-> { M } ) ) |
15 |
13 14
|
ax-mp |
|- ( { <. 1 , M >. } : ( 1 ... 1 ) -1-1-onto-> { M } <-> { <. 1 , M >. } : { 1 } -1-1-onto-> { M } ) |
16 |
11 15
|
sylibr |
|- ( ( 1 e. ZZ /\ M e. V ) -> { <. 1 , M >. } : ( 1 ... 1 ) -1-1-onto-> { M } ) |
17 |
10 16
|
mpan |
|- ( M e. V -> { <. 1 , M >. } : ( 1 ... 1 ) -1-1-onto-> { M } ) |
18 |
17
|
adantr |
|- ( ( M e. V /\ B e. CC ) -> { <. 1 , M >. } : ( 1 ... 1 ) -1-1-onto-> { M } ) |
19 |
|
velsn |
|- ( m e. { M } <-> m = M ) |
20 |
|
csbeq1 |
|- ( m = M -> [_ m / k ]_ A = [_ M / k ]_ A ) |
21 |
1
|
a1i |
|- ( M e. V -> F/_ k B ) |
22 |
21 2
|
csbiegf |
|- ( M e. V -> [_ M / k ]_ A = B ) |
23 |
22
|
adantr |
|- ( ( M e. V /\ B e. CC ) -> [_ M / k ]_ A = B ) |
24 |
20 23
|
sylan9eqr |
|- ( ( ( M e. V /\ B e. CC ) /\ m = M ) -> [_ m / k ]_ A = B ) |
25 |
19 24
|
sylan2b |
|- ( ( ( M e. V /\ B e. CC ) /\ m e. { M } ) -> [_ m / k ]_ A = B ) |
26 |
|
simplr |
|- ( ( ( M e. V /\ B e. CC ) /\ m e. { M } ) -> B e. CC ) |
27 |
25 26
|
eqeltrd |
|- ( ( ( M e. V /\ B e. CC ) /\ m e. { M } ) -> [_ m / k ]_ A e. CC ) |
28 |
13
|
eleq2i |
|- ( n e. ( 1 ... 1 ) <-> n e. { 1 } ) |
29 |
|
velsn |
|- ( n e. { 1 } <-> n = 1 ) |
30 |
28 29
|
bitri |
|- ( n e. ( 1 ... 1 ) <-> n = 1 ) |
31 |
|
fvsng |
|- ( ( 1 e. ZZ /\ M e. V ) -> ( { <. 1 , M >. } ` 1 ) = M ) |
32 |
10 31
|
mpan |
|- ( M e. V -> ( { <. 1 , M >. } ` 1 ) = M ) |
33 |
32
|
adantr |
|- ( ( M e. V /\ B e. CC ) -> ( { <. 1 , M >. } ` 1 ) = M ) |
34 |
33
|
csbeq1d |
|- ( ( M e. V /\ B e. CC ) -> [_ ( { <. 1 , M >. } ` 1 ) / k ]_ A = [_ M / k ]_ A ) |
35 |
|
simpr |
|- ( ( M e. V /\ B e. CC ) -> B e. CC ) |
36 |
|
fvsng |
|- ( ( 1 e. ZZ /\ B e. CC ) -> ( { <. 1 , B >. } ` 1 ) = B ) |
37 |
10 35 36
|
sylancr |
|- ( ( M e. V /\ B e. CC ) -> ( { <. 1 , B >. } ` 1 ) = B ) |
38 |
23 34 37
|
3eqtr4rd |
|- ( ( M e. V /\ B e. CC ) -> ( { <. 1 , B >. } ` 1 ) = [_ ( { <. 1 , M >. } ` 1 ) / k ]_ A ) |
39 |
|
fveq2 |
|- ( n = 1 -> ( { <. 1 , B >. } ` n ) = ( { <. 1 , B >. } ` 1 ) ) |
40 |
|
fveq2 |
|- ( n = 1 -> ( { <. 1 , M >. } ` n ) = ( { <. 1 , M >. } ` 1 ) ) |
41 |
40
|
csbeq1d |
|- ( n = 1 -> [_ ( { <. 1 , M >. } ` n ) / k ]_ A = [_ ( { <. 1 , M >. } ` 1 ) / k ]_ A ) |
42 |
39 41
|
eqeq12d |
|- ( n = 1 -> ( ( { <. 1 , B >. } ` n ) = [_ ( { <. 1 , M >. } ` n ) / k ]_ A <-> ( { <. 1 , B >. } ` 1 ) = [_ ( { <. 1 , M >. } ` 1 ) / k ]_ A ) ) |
43 |
38 42
|
syl5ibrcom |
|- ( ( M e. V /\ B e. CC ) -> ( n = 1 -> ( { <. 1 , B >. } ` n ) = [_ ( { <. 1 , M >. } ` n ) / k ]_ A ) ) |
44 |
43
|
imp |
|- ( ( ( M e. V /\ B e. CC ) /\ n = 1 ) -> ( { <. 1 , B >. } ` n ) = [_ ( { <. 1 , M >. } ` n ) / k ]_ A ) |
45 |
30 44
|
sylan2b |
|- ( ( ( M e. V /\ B e. CC ) /\ n e. ( 1 ... 1 ) ) -> ( { <. 1 , B >. } ` n ) = [_ ( { <. 1 , M >. } ` n ) / k ]_ A ) |
46 |
7 9 18 27 45
|
fprod |
|- ( ( M e. V /\ B e. CC ) -> prod_ m e. { M } [_ m / k ]_ A = ( seq 1 ( x. , { <. 1 , B >. } ) ` 1 ) ) |
47 |
6 46
|
eqtrid |
|- ( ( M e. V /\ B e. CC ) -> prod_ k e. { M } A = ( seq 1 ( x. , { <. 1 , B >. } ) ` 1 ) ) |
48 |
10 37
|
seq1i |
|- ( ( M e. V /\ B e. CC ) -> ( seq 1 ( x. , { <. 1 , B >. } ) ` 1 ) = B ) |
49 |
47 48
|
eqtrd |
|- ( ( M e. V /\ B e. CC ) -> prod_ k e. { M } A = B ) |