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Theorem srgbinom 19217
Description: The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)) (generalization of binom 15177). (Contributed by AV, 24-Aug-2019.)
Hypotheses
Ref Expression
srgbinom.s 𝑆 = (Base‘𝑅)
srgbinom.m × = (.r𝑅)
srgbinom.t · = (.g𝑅)
srgbinom.a + = (+g𝑅)
srgbinom.g 𝐺 = (mulGrp‘𝑅)
srgbinom.e = (.g𝐺)
Assertion
Ref Expression
srgbinom (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝑁   𝑅,𝑘   𝑆,𝑘   · ,𝑘   ,𝑘   × ,𝑘   + ,𝑘
Allowed substitution hint:   𝐺(𝑘)

Proof of Theorem srgbinom
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7158 . . . . . . 7 (𝑥 = 0 → (𝑥 (𝐴 + 𝐵)) = (0 (𝐴 + 𝐵)))
2 oveq2 7159 . . . . . . . . 9 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 7158 . . . . . . . . . 10 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 7158 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq1d 7166 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑥𝑘) 𝐴) = ((0 − 𝑘) 𝐴))
65oveq1d 7166 . . . . . . . . . 10 (𝑥 = 0 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))
73, 6oveq12d 7169 . . . . . . . . 9 (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))
82, 7mpteq12dv 5147 . . . . . . . 8 (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))
98oveq2d 7167 . . . . . . 7 (𝑥 = 0 → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))
101, 9eqeq12d 2841 . . . . . 6 (𝑥 = 0 → ((𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) ↔ (0 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))))
1110imbi2d 342 . . . . 5 (𝑥 = 0 → (((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))))
12 oveq1 7158 . . . . . . 7 (𝑥 = 𝑛 → (𝑥 (𝐴 + 𝐵)) = (𝑛 (𝐴 + 𝐵)))
13 oveq2 7159 . . . . . . . . 9 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 7158 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 7158 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq1d 7166 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑥𝑘) 𝐴) = ((𝑛𝑘) 𝐴))
1716oveq1d 7166 . . . . . . . . . 10 (𝑥 = 𝑛 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))
1814, 17oveq12d 7169 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))
1913, 18mpteq12dv 5147 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))
2019oveq2d 7167 . . . . . . 7 (𝑥 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))))
2112, 20eqeq12d 2841 . . . . . 6 (𝑥 = 𝑛 → ((𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) ↔ (𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))))
2221imbi2d 342 . . . . 5 (𝑥 = 𝑛 → (((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))))))
23 oveq1 7158 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑥 (𝐴 + 𝐵)) = ((𝑛 + 1) (𝐴 + 𝐵)))
24 oveq2 7159 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 7158 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 7158 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq1d 7166 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑥𝑘) 𝐴) = (((𝑛 + 1) − 𝑘) 𝐴))
2827oveq1d 7166 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))
2925, 28oveq12d 7169 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))
3024, 29mpteq12dv 5147 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))))
3130oveq2d 7167 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))
3223, 31eqeq12d 2841 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) ↔ ((𝑛 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))))))
3332imbi2d 342 . . . . 5 (𝑥 = (𝑛 + 1) → (((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((𝑛 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))))
34 oveq1 7158 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 (𝐴 + 𝐵)) = (𝑁 (𝐴 + 𝐵)))
35 oveq2 7159 . . . . . . . . 9 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 7158 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 7158 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq1d 7166 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝑥𝑘) 𝐴) = ((𝑁𝑘) 𝐴))
3938oveq1d 7166 . . . . . . . . . 10 (𝑥 = 𝑁 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))
4036, 39oveq12d 7169 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))
4135, 40mpteq12dv 5147 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))
4241oveq2d 7167 . . . . . . 7 (𝑥 = 𝑁 → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))
4334, 42eqeq12d 2841 . . . . . 6 (𝑥 = 𝑁 → ((𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))))) ↔ (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
4443imbi2d 342 . . . . 5 (𝑥 = 𝑁 → (((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))))
45 simpr1 1188 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴𝑆)
46 srgbinom.g . . . . . . . . . . . 12 𝐺 = (mulGrp‘𝑅)
47 srgbinom.s . . . . . . . . . . . 12 𝑆 = (Base‘𝑅)
4846, 47mgpbas 19167 . . . . . . . . . . 11 𝑆 = (Base‘𝐺)
4945, 48syl6eleq 2927 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50 eqid 2825 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
51 eqid 2825 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
52 srgbinom.e . . . . . . . . . . 11 = (.g𝐺)
5350, 51, 52mulg0 18163 . . . . . . . . . 10 (𝐴 ∈ (Base‘𝐺) → (0 𝐴) = (0g𝐺))
5449, 53syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐴) = (0g𝐺))
55 simpr2 1189 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵𝑆)
5655, 48syl6eleq 2927 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
5750, 51, 52mulg0 18163 . . . . . . . . . 10 (𝐵 ∈ (Base‘𝐺) → (0 𝐵) = (0g𝐺))
5856, 57syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐵) = (0g𝐺))
5954, 58oveq12d 7169 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 𝐴) × (0 𝐵)) = ((0g𝐺) × (0g𝐺)))
6059oveq2d 7167 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (1 · ((0g𝐺) × (0g𝐺))))
61 eqid 2825 . . . . . . . . . . . . . 14 (1r𝑅) = (1r𝑅)
6247, 61srgidcl 19190 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → (1r𝑅) ∈ 𝑆)
6362ancli 549 . . . . . . . . . . . 12 (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
6463adantr 481 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
65 srgbinom.m . . . . . . . . . . . 12 × = (.r𝑅)
6647, 65, 61srglidm 19193 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6764, 66syl 17 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6867oveq2d 7167 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · (1r𝑅)))
69 eqid 2825 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
7069, 61srgidcl 19190 . . . . . . . . . . 11 (𝑅 ∈ SRing → (1r𝑅) ∈ (Base‘𝑅))
71 srgbinom.t . . . . . . . . . . . 12 · = (.g𝑅)
7269, 71mulg1 18167 . . . . . . . . . . 11 ((1r𝑅) ∈ (Base‘𝑅) → (1 · (1r𝑅)) = (1r𝑅))
7370, 72syl 17 . . . . . . . . . 10 (𝑅 ∈ SRing → (1 · (1r𝑅)) = (1r𝑅))
7473adantr 481 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r𝑅)) = (1r𝑅))
7568, 74eqtrd 2860 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅))
7646, 61ringidval 19175 . . . . . . . . 9 (1r𝑅) = (0g𝐺)
77 id 22 . . . . . . . . . . . 12 ((1r𝑅) = (0g𝐺) → (1r𝑅) = (0g𝐺))
7877, 77oveq12d 7169 . . . . . . . . . . 11 ((1r𝑅) = (0g𝐺) → ((1r𝑅) × (1r𝑅)) = ((0g𝐺) × (0g𝐺)))
7978oveq2d 7167 . . . . . . . . . 10 ((1r𝑅) = (0g𝐺) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · ((0g𝐺) × (0g𝐺))))
8079, 77eqeq12d 2841 . . . . . . . . 9 ((1r𝑅) = (0g𝐺) → ((1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅) ↔ (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺)))
8176, 80ax-mp 5 . . . . . . . 8 ((1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅) ↔ (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺))
8275, 81sylib 219 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺))
8360, 82eqtrd 2860 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (0g𝐺))
84 fz0sn 13000 . . . . . . . . . 10 (0...0) = {0}
8584a1i 11 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0...0) = {0})
8685mpteq1d 5151 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))
8786oveq2d 7167 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))) = (𝑅 Σg (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))
88 srgmnd 19181 . . . . . . . . 9 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
8988adantr 481 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
90 c0ex 10627 . . . . . . . . 9 0 ∈ V
9190a1i 11 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
9276, 62eqeltrrid 2922 . . . . . . . . . 10 (𝑅 ∈ SRing → (0g𝐺) ∈ 𝑆)
9392adantr 481 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g𝐺) ∈ 𝑆)
9483, 93eqeltrd 2917 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) ∈ 𝑆)
95 oveq2 7159 . . . . . . . . . . 11 (𝑘 = 0 → (0C𝑘) = (0C0))
96 0nn0 11904 . . . . . . . . . . . 12 0 ∈ ℕ0
97 bcn0 13663 . . . . . . . . . . . 12 (0 ∈ ℕ0 → (0C0) = 1)
9896, 97ax-mp 5 . . . . . . . . . . 11 (0C0) = 1
9995, 98syl6eq 2876 . . . . . . . . . 10 (𝑘 = 0 → (0C𝑘) = 1)
100 oveq2 7159 . . . . . . . . . . . . 13 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
101 0m0e0 11749 . . . . . . . . . . . . 13 (0 − 0) = 0
102100, 101syl6eq 2876 . . . . . . . . . . . 12 (𝑘 = 0 → (0 − 𝑘) = 0)
103102oveq1d 7166 . . . . . . . . . . 11 (𝑘 = 0 → ((0 − 𝑘) 𝐴) = (0 𝐴))
104 oveq1 7158 . . . . . . . . . . 11 (𝑘 = 0 → (𝑘 𝐵) = (0 𝐵))
105103, 104oveq12d 7169 . . . . . . . . . 10 (𝑘 = 0 → (((0 − 𝑘) 𝐴) × (𝑘 𝐵)) = ((0 𝐴) × (0 𝐵)))
10699, 105oveq12d 7169 . . . . . . . . 9 (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))) = (1 · ((0 𝐴) × (0 𝐵))))
10747, 106gsumsn 18996 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 0 ∈ V ∧ (1 · ((0 𝐴) × (0 𝐵))) ∈ 𝑆) → (𝑅 Σg (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))) = (1 · ((0 𝐴) × (0 𝐵))))
10889, 91, 94, 107syl3anc 1365 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))) = (1 · ((0 𝐴) × (0 𝐵))))
10987, 108eqtrd 2860 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))) = (1 · ((0 𝐴) × (0 𝐵))))
110 srgbinom.a . . . . . . . . . 10 + = (+g𝑅)
11147, 110mndcl 17909 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑆𝐵𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
11289, 45, 55, 111syl3anc 1365 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
113112, 48syl6eleq 2927 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
11450, 51, 52mulg0 18163 . . . . . . 7 ((𝐴 + 𝐵) ∈ (Base‘𝐺) → (0 (𝐴 + 𝐵)) = (0g𝐺))
115113, 114syl 17 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (0g𝐺))
11683, 109, 1153eqtr4rd 2871 . . . . 5 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))
117 simprl 767 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑅 ∈ SRing)
11845adantl 482 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴𝑆)
11955adantl 482 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵𝑆)
120 simprr3 1217 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
121 simpl 483 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑛 ∈ ℕ0)
122 id 22 . . . . . . . 8 ((𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))) → (𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))))
12347, 65, 71, 110, 46, 52, 117, 118, 119, 120, 121, 122srgbinomlem 19216 . . . . . . 7 (((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) ∧ (𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))) → ((𝑛 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))
124123exp31 420 . . . . . 6 (𝑛 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))) → ((𝑛 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))))
125124a2d 29 . . . . 5 (𝑛 ∈ ℕ0 → (((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑛 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))) → ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((𝑛 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))))
12611, 22, 33, 44, 116, 125nn0ind 12069 . . . 4 (𝑁 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
127126expd 416 . . 3 (𝑁 ∈ ℕ0 → (𝑅 ∈ SRing → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))))
128127impcom 408 . 2 ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
129128imp 407 1 (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3499  {csn 4563  cmpt 5142  cfv 6351  (class class class)co 7151  0cc0 10529  1c1 10530   + caddc 10532  cmin 10862  0cn0 11889  ...cfz 12885  Ccbc 13655  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  0gc0g 16705   Σg cgsu 16706  Mndcmnd 17902  .gcmg 18156  mulGrpcmgp 19161  1rcur 19173  SRingcsrg 19177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-fz 12886  df-fzo 13027  df-seq 13363  df-fac 13627  df-bc 13656  df-hash 13684  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-mulg 18157  df-cntz 18379  df-cmn 18830  df-mgp 19162  df-ur 19174  df-srg 19178
This theorem is referenced by:  csrgbinom  19218
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