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Theorem srgbinom 20048
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 15773). (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 7413 . . . . . . 7 (𝑥 = 0 → (𝑥 (𝐴 + 𝐵)) = (0 (𝐴 + 𝐵)))
2 oveq2 7414 . . . . . . . . 9 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 7413 . . . . . . . . . 10 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 7413 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq1d 7421 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑥𝑘) 𝐴) = ((0 − 𝑘) 𝐴))
65oveq1d 7421 . . . . . . . . . 10 (𝑥 = 0 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))
73, 6oveq12d 7424 . . . . . . . . 9 (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))
82, 7mpteq12dv 5239 . . . . . . . 8 (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))
98oveq2d 7422 . . . . . . 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 2749 . . . . . 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 341 . . . . 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 7413 . . . . . . 7 (𝑥 = 𝑛 → (𝑥 (𝐴 + 𝐵)) = (𝑛 (𝐴 + 𝐵)))
13 oveq2 7414 . . . . . . . . 9 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 7413 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 7413 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq1d 7421 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑥𝑘) 𝐴) = ((𝑛𝑘) 𝐴))
1716oveq1d 7421 . . . . . . . . . 10 (𝑥 = 𝑛 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))
1814, 17oveq12d 7424 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))
1913, 18mpteq12dv 5239 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))
2019oveq2d 7422 . . . . . . 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 2749 . . . . . 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 341 . . . . 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 7413 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑥 (𝐴 + 𝐵)) = ((𝑛 + 1) (𝐴 + 𝐵)))
24 oveq2 7414 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 7413 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 7413 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq1d 7421 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑥𝑘) 𝐴) = (((𝑛 + 1) − 𝑘) 𝐴))
2827oveq1d 7421 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))
2925, 28oveq12d 7424 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))
3024, 29mpteq12dv 5239 . . . . . . . 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 7422 . . . . . . 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 2749 . . . . . 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 341 . . . . 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 7413 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 (𝐴 + 𝐵)) = (𝑁 (𝐴 + 𝐵)))
35 oveq2 7414 . . . . . . . . 9 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 7413 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 7413 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq1d 7421 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝑥𝑘) 𝐴) = ((𝑁𝑘) 𝐴))
3938oveq1d 7421 . . . . . . . . . 10 (𝑥 = 𝑁 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))
4036, 39oveq12d 7424 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))
4135, 40mpteq12dv 5239 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))
4241oveq2d 7422 . . . . . . 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 2749 . . . . . 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 341 . . . . 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 1195 . . . . . . . . . . 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 19988 . . . . . . . . . . 11 𝑆 = (Base‘𝐺)
4945, 48eleqtrdi 2844 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50 eqid 2733 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
51 eqid 2733 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
52 srgbinom.e . . . . . . . . . . 11 = (.g𝐺)
5350, 51, 52mulg0 18952 . . . . . . . . . 10 (𝐴 ∈ (Base‘𝐺) → (0 𝐴) = (0g𝐺))
5449, 53syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐴) = (0g𝐺))
55 simpr2 1196 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵𝑆)
5655, 48eleqtrdi 2844 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
5750, 51, 52mulg0 18952 . . . . . . . . . 10 (𝐵 ∈ (Base‘𝐺) → (0 𝐵) = (0g𝐺))
5856, 57syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐵) = (0g𝐺))
5954, 58oveq12d 7424 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 𝐴) × (0 𝐵)) = ((0g𝐺) × (0g𝐺)))
6059oveq2d 7422 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (1 · ((0g𝐺) × (0g𝐺))))
61 eqid 2733 . . . . . . . . . . . . . 14 (1r𝑅) = (1r𝑅)
6247, 61srgidcl 20016 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → (1r𝑅) ∈ 𝑆)
6362ancli 550 . . . . . . . . . . . 12 (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
6463adantr 482 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
65 srgbinom.m . . . . . . . . . . . 12 × = (.r𝑅)
6647, 65, 61srglidm 20019 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6764, 66syl 17 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6867oveq2d 7422 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · (1r𝑅)))
69 eqid 2733 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
7069, 61srgidcl 20016 . . . . . . . . . . 11 (𝑅 ∈ SRing → (1r𝑅) ∈ (Base‘𝑅))
71 srgbinom.t . . . . . . . . . . . 12 · = (.g𝑅)
7269, 71mulg1 18956 . . . . . . . . . . 11 ((1r𝑅) ∈ (Base‘𝑅) → (1 · (1r𝑅)) = (1r𝑅))
7370, 72syl 17 . . . . . . . . . 10 (𝑅 ∈ SRing → (1 · (1r𝑅)) = (1r𝑅))
7473adantr 482 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r𝑅)) = (1r𝑅))
7568, 74eqtrd 2773 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅))
7646, 61ringidval 20001 . . . . . . . . 9 (1r𝑅) = (0g𝐺)
77 id 22 . . . . . . . . . . . 12 ((1r𝑅) = (0g𝐺) → (1r𝑅) = (0g𝐺))
7877, 77oveq12d 7424 . . . . . . . . . . 11 ((1r𝑅) = (0g𝐺) → ((1r𝑅) × (1r𝑅)) = ((0g𝐺) × (0g𝐺)))
7978oveq2d 7422 . . . . . . . . . 10 ((1r𝑅) = (0g𝐺) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · ((0g𝐺) × (0g𝐺))))
8079, 77eqeq12d 2749 . . . . . . . . 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 217 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺))
8360, 82eqtrd 2773 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (0g𝐺))
84 fz0sn 13598 . . . . . . . . . 10 (0...0) = {0}
8584a1i 11 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0...0) = {0})
8685mpteq1d 5243 . . . . . . . 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 7422 . . . . . . 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 20007 . . . . . . . . 9 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
8988adantr 482 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
90 c0ex 11205 . . . . . . . . 9 0 ∈ V
9190a1i 11 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
9276, 62eqeltrrid 2839 . . . . . . . . . 10 (𝑅 ∈ SRing → (0g𝐺) ∈ 𝑆)
9392adantr 482 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g𝐺) ∈ 𝑆)
9483, 93eqeltrd 2834 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) ∈ 𝑆)
95 oveq2 7414 . . . . . . . . . . 11 (𝑘 = 0 → (0C𝑘) = (0C0))
96 0nn0 12484 . . . . . . . . . . . 12 0 ∈ ℕ0
97 bcn0 14267 . . . . . . . . . . . 12 (0 ∈ ℕ0 → (0C0) = 1)
9896, 97ax-mp 5 . . . . . . . . . . 11 (0C0) = 1
9995, 98eqtrdi 2789 . . . . . . . . . 10 (𝑘 = 0 → (0C𝑘) = 1)
100 oveq2 7414 . . . . . . . . . . . . 13 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
101 0m0e0 12329 . . . . . . . . . . . . 13 (0 − 0) = 0
102100, 101eqtrdi 2789 . . . . . . . . . . . 12 (𝑘 = 0 → (0 − 𝑘) = 0)
103102oveq1d 7421 . . . . . . . . . . 11 (𝑘 = 0 → ((0 − 𝑘) 𝐴) = (0 𝐴))
104 oveq1 7413 . . . . . . . . . . 11 (𝑘 = 0 → (𝑘 𝐵) = (0 𝐵))
105103, 104oveq12d 7424 . . . . . . . . . 10 (𝑘 = 0 → (((0 − 𝑘) 𝐴) × (𝑘 𝐵)) = ((0 𝐴) × (0 𝐵)))
10699, 105oveq12d 7424 . . . . . . . . 9 (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))) = (1 · ((0 𝐴) × (0 𝐵))))
10747, 106gsumsn 19817 . . . . . . . 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 1372 . . . . . . 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 2773 . . . . . 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 18630 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑆𝐵𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
11289, 45, 55, 111syl3anc 1372 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
113112, 48eleqtrdi 2844 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
11450, 51, 52mulg0 18952 . . . . . . 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 2784 . . . . 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 483 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴𝑆)
11955adantl 483 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵𝑆)
120 simprr3 1224 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
121 simpl 484 . . . . . . . 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 20047 . . . . . . 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 421 . . . . . 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 12654 . . . 4 (𝑁 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
127126expd 417 . . 3 (𝑁 ∈ ℕ0 → (𝑅 ∈ SRing → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))))
128127impcom 409 . 2 ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
129128imp 408 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  {csn 4628  cmpt 5231  cfv 6541  (class class class)co 7406  0cc0 11107  1c1 11108   + caddc 11110  cmin 11441  0cn0 12469  ...cfz 13481  Ccbc 14259  Basecbs 17141  +gcplusg 17194  .rcmulr 17195  0gc0g 17382   Σg cgsu 17383  Mndcmnd 18622  .gcmg 18945  mulGrpcmgp 19982  1rcur 19999  SRingcsrg 20003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-supp 8144  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-seq 13964  df-fac 14231  df-bc 14260  df-hash 14288  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-0g 17384  df-gsum 17385  df-mre 17527  df-mrc 17528  df-acs 17530  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-mhm 18668  df-submnd 18669  df-mulg 18946  df-cntz 19176  df-cmn 19645  df-mgp 19983  df-ur 20000  df-srg 20004
This theorem is referenced by:  csrgbinom  20049
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