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Theorem srgbinom 19363
 Description: The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)) (generalization of binom 15233). (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 7157 . . . . . . 7 (𝑥 = 0 → (𝑥 (𝐴 + 𝐵)) = (0 (𝐴 + 𝐵)))
2 oveq2 7158 . . . . . . . . 9 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 7157 . . . . . . . . . 10 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 7157 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq1d 7165 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑥𝑘) 𝐴) = ((0 − 𝑘) 𝐴))
65oveq1d 7165 . . . . . . . . . 10 (𝑥 = 0 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))
73, 6oveq12d 7168 . . . . . . . . 9 (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))
82, 7mpteq12dv 5117 . . . . . . . 8 (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))
98oveq2d 7166 . . . . . . 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 2774 . . . . . 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 344 . . . . 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 7157 . . . . . . 7 (𝑥 = 𝑛 → (𝑥 (𝐴 + 𝐵)) = (𝑛 (𝐴 + 𝐵)))
13 oveq2 7158 . . . . . . . . 9 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 7157 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 7157 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq1d 7165 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑥𝑘) 𝐴) = ((𝑛𝑘) 𝐴))
1716oveq1d 7165 . . . . . . . . . 10 (𝑥 = 𝑛 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))
1814, 17oveq12d 7168 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))
1913, 18mpteq12dv 5117 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))
2019oveq2d 7166 . . . . . . 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 2774 . . . . . 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 344 . . . . 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 7157 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑥 (𝐴 + 𝐵)) = ((𝑛 + 1) (𝐴 + 𝐵)))
24 oveq2 7158 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 7157 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 7157 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq1d 7165 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑥𝑘) 𝐴) = (((𝑛 + 1) − 𝑘) 𝐴))
2827oveq1d 7165 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))
2925, 28oveq12d 7168 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))
3024, 29mpteq12dv 5117 . . . . . . . 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 7166 . . . . . . 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 2774 . . . . . 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 344 . . . . 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 7157 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 (𝐴 + 𝐵)) = (𝑁 (𝐴 + 𝐵)))
35 oveq2 7158 . . . . . . . . 9 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 7157 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 7157 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq1d 7165 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝑥𝑘) 𝐴) = ((𝑁𝑘) 𝐴))
3938oveq1d 7165 . . . . . . . . . 10 (𝑥 = 𝑁 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))
4036, 39oveq12d 7168 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))
4135, 40mpteq12dv 5117 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))
4241oveq2d 7166 . . . . . . 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 2774 . . . . . 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 344 . . . . 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 1191 . . . . . . . . . . 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 19313 . . . . . . . . . . 11 𝑆 = (Base‘𝐺)
4945, 48eleqtrdi 2862 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50 eqid 2758 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
51 eqid 2758 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
52 srgbinom.e . . . . . . . . . . 11 = (.g𝐺)
5350, 51, 52mulg0 18298 . . . . . . . . . 10 (𝐴 ∈ (Base‘𝐺) → (0 𝐴) = (0g𝐺))
5449, 53syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐴) = (0g𝐺))
55 simpr2 1192 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵𝑆)
5655, 48eleqtrdi 2862 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
5750, 51, 52mulg0 18298 . . . . . . . . . 10 (𝐵 ∈ (Base‘𝐺) → (0 𝐵) = (0g𝐺))
5856, 57syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐵) = (0g𝐺))
5954, 58oveq12d 7168 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 𝐴) × (0 𝐵)) = ((0g𝐺) × (0g𝐺)))
6059oveq2d 7166 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (1 · ((0g𝐺) × (0g𝐺))))
61 eqid 2758 . . . . . . . . . . . . . 14 (1r𝑅) = (1r𝑅)
6247, 61srgidcl 19336 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → (1r𝑅) ∈ 𝑆)
6362ancli 552 . . . . . . . . . . . 12 (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
6463adantr 484 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
65 srgbinom.m . . . . . . . . . . . 12 × = (.r𝑅)
6647, 65, 61srglidm 19339 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6764, 66syl 17 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6867oveq2d 7166 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · (1r𝑅)))
69 eqid 2758 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
7069, 61srgidcl 19336 . . . . . . . . . . 11 (𝑅 ∈ SRing → (1r𝑅) ∈ (Base‘𝑅))
71 srgbinom.t . . . . . . . . . . . 12 · = (.g𝑅)
7269, 71mulg1 18302 . . . . . . . . . . 11 ((1r𝑅) ∈ (Base‘𝑅) → (1 · (1r𝑅)) = (1r𝑅))
7370, 72syl 17 . . . . . . . . . 10 (𝑅 ∈ SRing → (1 · (1r𝑅)) = (1r𝑅))
7473adantr 484 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r𝑅)) = (1r𝑅))
7568, 74eqtrd 2793 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅))
7646, 61ringidval 19321 . . . . . . . . 9 (1r𝑅) = (0g𝐺)
77 id 22 . . . . . . . . . . . 12 ((1r𝑅) = (0g𝐺) → (1r𝑅) = (0g𝐺))
7877, 77oveq12d 7168 . . . . . . . . . . 11 ((1r𝑅) = (0g𝐺) → ((1r𝑅) × (1r𝑅)) = ((0g𝐺) × (0g𝐺)))
7978oveq2d 7166 . . . . . . . . . 10 ((1r𝑅) = (0g𝐺) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · ((0g𝐺) × (0g𝐺))))
8079, 77eqeq12d 2774 . . . . . . . . 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 221 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺))
8360, 82eqtrd 2793 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (0g𝐺))
84 fz0sn 13056 . . . . . . . . . 10 (0...0) = {0}
8584a1i 11 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0...0) = {0})
8685mpteq1d 5121 . . . . . . . 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 7166 . . . . . . 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 19327 . . . . . . . . 9 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
8988adantr 484 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
90 c0ex 10673 . . . . . . . . 9 0 ∈ V
9190a1i 11 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
9276, 62eqeltrrid 2857 . . . . . . . . . 10 (𝑅 ∈ SRing → (0g𝐺) ∈ 𝑆)
9392adantr 484 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g𝐺) ∈ 𝑆)
9483, 93eqeltrd 2852 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) ∈ 𝑆)
95 oveq2 7158 . . . . . . . . . . 11 (𝑘 = 0 → (0C𝑘) = (0C0))
96 0nn0 11949 . . . . . . . . . . . 12 0 ∈ ℕ0
97 bcn0 13720 . . . . . . . . . . . 12 (0 ∈ ℕ0 → (0C0) = 1)
9896, 97ax-mp 5 . . . . . . . . . . 11 (0C0) = 1
9995, 98eqtrdi 2809 . . . . . . . . . 10 (𝑘 = 0 → (0C𝑘) = 1)
100 oveq2 7158 . . . . . . . . . . . . 13 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
101 0m0e0 11794 . . . . . . . . . . . . 13 (0 − 0) = 0
102100, 101eqtrdi 2809 . . . . . . . . . . . 12 (𝑘 = 0 → (0 − 𝑘) = 0)
103102oveq1d 7165 . . . . . . . . . . 11 (𝑘 = 0 → ((0 − 𝑘) 𝐴) = (0 𝐴))
104 oveq1 7157 . . . . . . . . . . 11 (𝑘 = 0 → (𝑘 𝐵) = (0 𝐵))
105103, 104oveq12d 7168 . . . . . . . . . 10 (𝑘 = 0 → (((0 − 𝑘) 𝐴) × (𝑘 𝐵)) = ((0 𝐴) × (0 𝐵)))
10699, 105oveq12d 7168 . . . . . . . . 9 (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))) = (1 · ((0 𝐴) × (0 𝐵))))
10747, 106gsumsn 19142 . . . . . . . 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 1368 . . . . . . 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 2793 . . . . . 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 17985 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑆𝐵𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
11289, 45, 55, 111syl3anc 1368 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
113112, 48eleqtrdi 2862 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
11450, 51, 52mulg0 18298 . . . . . . 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 2804 . . . . 5 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))
117 simprl 770 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑅 ∈ SRing)
11845adantl 485 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴𝑆)
11955adantl 485 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵𝑆)
120 simprr3 1220 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
121 simpl 486 . . . . . . . 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 19362 . . . . . . 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 423 . . . . . 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 12116 . . . 4 (𝑁 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
127126expd 419 . . 3 (𝑁 ∈ ℕ0 → (𝑅 ∈ SRing → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))))
128127impcom 411 . 2 ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
129128imp 410 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4522   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576   + caddc 10578   − cmin 10908  ℕ0cn0 11934  ...cfz 12939  Ccbc 13712  Basecbs 16541  +gcplusg 16623  .rcmulr 16624  0gc0g 16771   Σg cgsu 16772  Mndcmnd 17977  .gcmg 18291  mulGrpcmgp 19307  1rcur 19319  SRingcsrg 19323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7405  df-om 7580  df-1st 7693  df-2nd 7694  df-supp 7836  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-fsupp 8867  df-oi 9007  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-n0 11935  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-fzo 13083  df-seq 13419  df-fac 13684  df-bc 13713  df-hash 13741  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-0g 16773  df-gsum 16774  df-mre 16915  df-mrc 16916  df-acs 16918  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-mhm 18022  df-submnd 18023  df-mulg 18292  df-cntz 18514  df-cmn 18975  df-mgp 19308  df-ur 19320  df-srg 19324 This theorem is referenced by:  csrgbinom  19364
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