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Metamath Proof Explorer

Theorem crngbinom

Description: The binomial theorem for commutative rings (special case of csrgbinom ): ( A + B ) ^ N is the sum from k = 0 to N of ( N _C k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ) . (Contributed by AV, 24-Aug-2019)

Ref Expression
Hypotheses crngbinom.s ⊢ 𝑆 = ( Base ‘ 𝑅 )
crngbinom.m ⊢ × = ( .r ‘ 𝑅 )
crngbinom.t ⊢ · = ( .g ‘ 𝑅 )
crngbinom.a ⊢ + = ( +g ‘ 𝑅 )
crngbinom.g ⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
crngbinom.e ⊢ ↑ = ( .g ‘ 𝐺 )
Assertion crngbinom ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝑁 ↑ ( 𝐴 + 𝐵 ) ) = ( 𝑅 Σg ( 𝑘 ∈ ( 0 ... 𝑁 ) ↦ ( ( 𝑁 C 𝑘 ) · ( ( ( 𝑁 − 𝑘 ) ↑ 𝐴 ) × ( 𝑘 ↑ 𝐵 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 crngbinom.s ⊢ 𝑆 = ( Base ‘ 𝑅 )
2 crngbinom.m ⊢ × = ( .r ‘ 𝑅 )
3 crngbinom.t ⊢ · = ( .g ‘ 𝑅 )
4 crngbinom.a ⊢ + = ( +g ‘ 𝑅 )
5 crngbinom.g ⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
6 crngbinom.e ⊢ ↑ = ( .g ‘ 𝐺 )
7 crngring ⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
8 ringsrg ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ SRing )
9 7 8 syl ⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ SRing )
10 9 adantr ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ) → 𝑅 ∈ SRing )
11 5 crngmgp ⊢ ( 𝑅 ∈ CRing → 𝐺 ∈ CMnd )
12 11 adantr ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ) → 𝐺 ∈ CMnd )
13 simpr ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 )
14 10 12 13 3jca ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ) → ( 𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0 ) )
15 1 2 3 4 5 6 csrgbinom ⊢ ( ( ( 𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝑁 ↑ ( 𝐴 + 𝐵 ) ) = ( 𝑅 Σg ( 𝑘 ∈ ( 0 ... 𝑁 ) ↦ ( ( 𝑁 C 𝑘 ) · ( ( ( 𝑁 − 𝑘 ) ↑ 𝐴 ) × ( 𝑘 ↑ 𝐵 ) ) ) ) ) )
16 14 15 sylan ⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝑁 ↑ ( 𝐴 + 𝐵 ) ) = ( 𝑅 Σg ( 𝑘 ∈ ( 0 ... 𝑁 ) ↦ ( ( 𝑁 C 𝑘 ) · ( ( ( 𝑁 − 𝑘 ) ↑ 𝐴 ) × ( 𝑘 ↑ 𝐵 ) ) ) ) ) )