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Metamath Proof Explorer

Theorem fprodm1s

Description: Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017)

Ref Expression
Hypotheses fprodm1s.1 
|- ( ph -> N e. ( ZZ>= ` M ) )
fprodm1s.2 
|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
Assertion fprodm1s 
|- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) )

Proof

Step Hyp Ref Expression
1 fprodm1s.1 
 |-  ( ph -> N e. ( ZZ>= ` M ) )
2 fprodm1s.2 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
3 2 ralrimiva 
 |-  ( ph -> A. k e. ( M ... N ) A e. CC )
4 nfcsb1v 
 |-  F/_ k [_ m / k ]_ A
5 4 nfel1 
 |-  F/ k [_ m / k ]_ A e. CC
6 csbeq1a 
 |-  ( k = m -> A = [_ m / k ]_ A )
7 6 eleq1d 
 |-  ( k = m -> ( A e. CC <-> [_ m / k ]_ A e. CC ) )
8 5 7 rspc 
 |-  ( m e. ( M ... N ) -> ( A. k e. ( M ... N ) A e. CC -> [_ m / k ]_ A e. CC ) )
9 3 8 mpan9 
 |-  ( ( ph /\ m e. ( M ... N ) ) -> [_ m / k ]_ A e. CC )
10 csbeq1 
 |-  ( m = N -> [_ m / k ]_ A = [_ N / k ]_ A )
11 1 9 10 fprodm1 
 |-  ( ph -> prod_ m e. ( M ... N ) [_ m / k ]_ A = ( prod_ m e. ( M ... ( N - 1 ) ) [_ m / k ]_ A x. [_ N / k ]_ A ) )
12 nfcv 
 |-  F/_ m A
13 12 4 6 cbvprodi 
 |-  prod_ k e. ( M ... N ) A = prod_ m e. ( M ... N ) [_ m / k ]_ A
14 12 4 6 cbvprodi 
 |-  prod_ k e. ( M ... ( N - 1 ) ) A = prod_ m e. ( M ... ( N - 1 ) ) [_ m / k ]_ A
15 14 oveq1i 
 |-  ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) = ( prod_ m e. ( M ... ( N - 1 ) ) [_ m / k ]_ A x. [_ N / k ]_ A )
16 11 13 15 3eqtr4g 
 |-  ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) )