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Theorem srgbinom 18368
 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 14401). (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 6556 . . . . . . 7 (𝑥 = 0 → (𝑥 (𝐴 + 𝐵)) = (0 (𝐴 + 𝐵)))
2 oveq2 6557 . . . . . . . . 9 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 6556 . . . . . . . . . 10 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq1d 6564 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑥𝑘) 𝐴) = ((0 − 𝑘) 𝐴))
65oveq1d 6564 . . . . . . . . . 10 (𝑥 = 0 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))
73, 6oveq12d 6567 . . . . . . . . 9 (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))
82, 7mpteq12dv 4663 . . . . . . . 8 (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))
98oveq2d 6565 . . . . . . 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 2625 . . . . . 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 329 . . . . 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 6556 . . . . . . 7 (𝑥 = 𝑛 → (𝑥 (𝐴 + 𝐵)) = (𝑛 (𝐴 + 𝐵)))
13 oveq2 6557 . . . . . . . . 9 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 6556 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq1d 6564 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑥𝑘) 𝐴) = ((𝑛𝑘) 𝐴))
1716oveq1d 6564 . . . . . . . . . 10 (𝑥 = 𝑛 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))
1814, 17oveq12d 6567 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))
1913, 18mpteq12dv 4663 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))
2019oveq2d 6565 . . . . . . 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 2625 . . . . . 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 329 . . . . 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 6556 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑥 (𝐴 + 𝐵)) = ((𝑛 + 1) (𝐴 + 𝐵)))
24 oveq2 6557 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 6556 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq1d 6564 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑥𝑘) 𝐴) = (((𝑛 + 1) − 𝑘) 𝐴))
2827oveq1d 6564 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))
2925, 28oveq12d 6567 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))
3024, 29mpteq12dv 4663 . . . . . . . 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 6565 . . . . . . 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 2625 . . . . . 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 329 . . . . 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 6556 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 (𝐴 + 𝐵)) = (𝑁 (𝐴 + 𝐵)))
35 oveq2 6557 . . . . . . . . 9 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 6556 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq1d 6564 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝑥𝑘) 𝐴) = ((𝑁𝑘) 𝐴))
3938oveq1d 6564 . . . . . . . . . 10 (𝑥 = 𝑁 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))
4036, 39oveq12d 6567 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))
4135, 40mpteq12dv 4663 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))
4241oveq2d 6565 . . . . . . 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 2625 . . . . . 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 329 . . . . 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 1060 . . . . . . . . . . 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 18318 . . . . . . . . . . 11 𝑆 = (Base‘𝐺)
4945, 48syl6eleq 2698 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50 eqid 2610 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
51 eqid 2610 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
52 srgbinom.e . . . . . . . . . . 11 = (.g𝐺)
5350, 51, 52mulg0 17369 . . . . . . . . . 10 (𝐴 ∈ (Base‘𝐺) → (0 𝐴) = (0g𝐺))
5449, 53syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐴) = (0g𝐺))
55 simpr2 1061 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵𝑆)
5655, 48syl6eleq 2698 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
5750, 51, 52mulg0 17369 . . . . . . . . . 10 (𝐵 ∈ (Base‘𝐺) → (0 𝐵) = (0g𝐺))
5856, 57syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐵) = (0g𝐺))
5954, 58oveq12d 6567 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 𝐴) × (0 𝐵)) = ((0g𝐺) × (0g𝐺)))
6059oveq2d 6565 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (1 · ((0g𝐺) × (0g𝐺))))
61 eqid 2610 . . . . . . . . . . . . . 14 (1r𝑅) = (1r𝑅)
6247, 61srgidcl 18341 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → (1r𝑅) ∈ 𝑆)
6362ancli 572 . . . . . . . . . . . 12 (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
6463adantr 480 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
65 srgbinom.m . . . . . . . . . . . 12 × = (.r𝑅)
6647, 65, 61srglidm 18344 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6764, 66syl 17 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6867oveq2d 6565 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · (1r𝑅)))
69 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
7069, 61srgidcl 18341 . . . . . . . . . . 11 (𝑅 ∈ SRing → (1r𝑅) ∈ (Base‘𝑅))
71 srgbinom.t . . . . . . . . . . . 12 · = (.g𝑅)
7269, 71mulg1 17371 . . . . . . . . . . 11 ((1r𝑅) ∈ (Base‘𝑅) → (1 · (1r𝑅)) = (1r𝑅))
7370, 72syl 17 . . . . . . . . . 10 (𝑅 ∈ SRing → (1 · (1r𝑅)) = (1r𝑅))
7473adantr 480 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r𝑅)) = (1r𝑅))
7568, 74eqtrd 2644 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅))
7646, 61ringidval 18326 . . . . . . . . 9 (1r𝑅) = (0g𝐺)
77 id 22 . . . . . . . . . . . 12 ((1r𝑅) = (0g𝐺) → (1r𝑅) = (0g𝐺))
7877, 77oveq12d 6567 . . . . . . . . . . 11 ((1r𝑅) = (0g𝐺) → ((1r𝑅) × (1r𝑅)) = ((0g𝐺) × (0g𝐺)))
7978oveq2d 6565 . . . . . . . . . 10 ((1r𝑅) = (0g𝐺) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · ((0g𝐺) × (0g𝐺))))
8079, 77eqeq12d 2625 . . . . . . . . 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 207 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺))
8360, 82eqtrd 2644 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (0g𝐺))
84 0z 11265 . . . . . . . . . . 11 0 ∈ ℤ
85 fzsn 12254 . . . . . . . . . . 11 (0 ∈ ℤ → (0...0) = {0})
8684, 85ax-mp 5 . . . . . . . . . 10 (0...0) = {0}
8786a1i 11 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0...0) = {0})
8887mpteq1d 4666 . . . . . . . 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)))))
8988oveq2d 6565 . . . . . . 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))))))
90 srgmnd 18332 . . . . . . . . 9 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
9190adantr 480 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
92 c0ex 9913 . . . . . . . . 9 0 ∈ V
9392a1i 11 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
9476, 62syl5eqelr 2693 . . . . . . . . . 10 (𝑅 ∈ SRing → (0g𝐺) ∈ 𝑆)
9594adantr 480 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g𝐺) ∈ 𝑆)
9683, 95eqeltrd 2688 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) ∈ 𝑆)
97 oveq2 6557 . . . . . . . . . . 11 (𝑘 = 0 → (0C𝑘) = (0C0))
98 0nn0 11184 . . . . . . . . . . . 12 0 ∈ ℕ0
99 bcn0 12959 . . . . . . . . . . . 12 (0 ∈ ℕ0 → (0C0) = 1)
10098, 99ax-mp 5 . . . . . . . . . . 11 (0C0) = 1
10197, 100syl6eq 2660 . . . . . . . . . 10 (𝑘 = 0 → (0C𝑘) = 1)
102 oveq2 6557 . . . . . . . . . . . . 13 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
103 0m0e0 11007 . . . . . . . . . . . . 13 (0 − 0) = 0
104102, 103syl6eq 2660 . . . . . . . . . . . 12 (𝑘 = 0 → (0 − 𝑘) = 0)
105104oveq1d 6564 . . . . . . . . . . 11 (𝑘 = 0 → ((0 − 𝑘) 𝐴) = (0 𝐴))
106 oveq1 6556 . . . . . . . . . . 11 (𝑘 = 0 → (𝑘 𝐵) = (0 𝐵))
107105, 106oveq12d 6567 . . . . . . . . . 10 (𝑘 = 0 → (((0 − 𝑘) 𝐴) × (𝑘 𝐵)) = ((0 𝐴) × (0 𝐵)))
108101, 107oveq12d 6567 . . . . . . . . 9 (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))) = (1 · ((0 𝐴) × (0 𝐵))))
10947, 108gsumsn 18177 . . . . . . . 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))))
11091, 93, 96, 109syl3anc 1318 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))) = (1 · ((0 𝐴) × (0 𝐵))))
11189, 110eqtrd 2644 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))) = (1 · ((0 𝐴) × (0 𝐵))))
112 srgbinom.a . . . . . . . . . 10 + = (+g𝑅)
11347, 112mndcl 17124 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑆𝐵𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
11491, 45, 55, 113syl3anc 1318 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
115114, 48syl6eleq 2698 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
11650, 51, 52mulg0 17369 . . . . . . 7 ((𝐴 + 𝐵) ∈ (Base‘𝐺) → (0 (𝐴 + 𝐵)) = (0g𝐺))
117115, 116syl 17 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (0g𝐺))
11883, 111, 1173eqtr4rd 2655 . . . . 5 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))
119 simprl 790 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑅 ∈ SRing)
12045adantl 481 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴𝑆)
12155adantl 481 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵𝑆)
122 simprr3 1104 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
123 simpl 472 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑛 ∈ ℕ0)
124 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))))))
12547, 65, 71, 112, 46, 52, 119, 120, 121, 122, 123, 124srgbinomlem 18367 . . . . . . 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))))))
126125exp31 628 . . . . . 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))))))))
127126a2d 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))))))))
12811, 22, 33, 44, 118, 127nn0ind 11348 . . . 4 (𝑁 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
129128expd 451 . . 3 (𝑁 ∈ ℕ0 → (𝑅 ∈ SRing → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))))
130129impcom 445 . 2 ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → ((𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))))
131130imp 444 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  Ccbc 12951  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  .gcmg 17363  mulGrpcmgp 18312  1rcur 18324  SRingcsrg 18328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-fac 12923  df-bc 12952  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-mgp 18313  df-ur 18325  df-srg 18329 This theorem is referenced by:  csrgbinom  18369
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