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Metamath Proof Explorer

Theorem csrgbinom

Description: The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019)

Ref Expression
Hypotheses srgbinom.s 
|- S = ( Base ` R )
srgbinom.m 
|- .X. = ( .r ` R )
srgbinom.t 
|- .x. = ( .g ` R )
srgbinom.a 
|- .+ = ( +g ` R )
srgbinom.g 
|- G = ( mulGrp ` R )
srgbinom.e 
|- .^ = ( .g ` G )
Assertion csrgbinom 
|- ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> ( N .^ ( A .+ B ) ) = ( R gsum ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ A ) .X. ( k .^ B ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 srgbinom.s 
 |-  S = ( Base ` R )
2 srgbinom.m 
 |-  .X. = ( .r ` R )
3 srgbinom.t 
 |-  .x. = ( .g ` R )
4 srgbinom.a 
 |-  .+ = ( +g ` R )
5 srgbinom.g 
 |-  G = ( mulGrp ` R )
6 srgbinom.e 
 |-  .^ = ( .g ` G )
7 3simpb 
 |-  ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) -> ( R e. SRing /\ N e. NN0 ) )
8 7 adantr 
 |-  ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> ( R e. SRing /\ N e. NN0 ) )
9 simprl 
 |-  ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> A e. S )
10 simprr 
 |-  ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> B e. S )
11 simpl2 
 |-  ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> G e. CMnd )
12 5 1 mgpbas 
 |-  S = ( Base ` G )
13 5 2 mgpplusg 
 |-  .X. = ( +g ` G )
14 12 13 cmncom 
 |-  ( ( G e. CMnd /\ A e. S /\ B e. S ) -> ( A .X. B ) = ( B .X. A ) )
15 11 9 10 14 syl3anc 
 |-  ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> ( A .X. B ) = ( B .X. A ) )
16 1 2 3 4 5 6 srgbinom 
 |-  ( ( ( R e. SRing /\ N e. NN0 ) /\ ( A e. S /\ B e. S /\ ( A .X. B ) = ( B .X. A ) ) ) -> ( N .^ ( A .+ B ) ) = ( R gsum ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ A ) .X. ( k .^ B ) ) ) ) ) )
17 8 9 10 15 16 syl13anc 
 |-  ( ( ( R e. SRing /\ G e. CMnd /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> ( N .^ ( A .+ B ) ) = ( R gsum ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ A ) .X. ( k .^ B ) ) ) ) ) )