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Theorem srgbinom 18737
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 14753). (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 6812 . . . . . . 7 (𝑥 = 0 → (𝑥 (𝐴 + 𝐵)) = (0 (𝐴 + 𝐵)))
2 oveq2 6813 . . . . . . . . 9 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 6812 . . . . . . . . . 10 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 6812 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq1d 6820 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑥𝑘) 𝐴) = ((0 − 𝑘) 𝐴))
65oveq1d 6820 . . . . . . . . . 10 (𝑥 = 0 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))
73, 6oveq12d 6823 . . . . . . . . 9 (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))
82, 7mpteq12dv 4877 . . . . . . . 8 (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵)))))
98oveq2d 6821 . . . . . . 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 2767 . . . . . 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 6812 . . . . . . 7 (𝑥 = 𝑛 → (𝑥 (𝐴 + 𝐵)) = (𝑛 (𝐴 + 𝐵)))
13 oveq2 6813 . . . . . . . . 9 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 6812 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 6812 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq1d 6820 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑥𝑘) 𝐴) = ((𝑛𝑘) 𝐴))
1716oveq1d 6820 . . . . . . . . . 10 (𝑥 = 𝑛 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))
1814, 17oveq12d 6823 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵))))
1913, 18mpteq12dv 4877 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛𝑘) 𝐴) × (𝑘 𝐵)))))
2019oveq2d 6821 . . . . . . 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 2767 . . . . . 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 6812 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑥 (𝐴 + 𝐵)) = ((𝑛 + 1) (𝐴 + 𝐵)))
24 oveq2 6813 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 6812 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 6812 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq1d 6820 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑥𝑘) 𝐴) = (((𝑛 + 1) − 𝑘) 𝐴))
2827oveq1d 6820 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵)))
2925, 28oveq12d 6823 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))
3024, 29mpteq12dv 4877 . . . . . . . 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 6821 . . . . . . 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 2767 . . . . . 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 6812 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 (𝐴 + 𝐵)) = (𝑁 (𝐴 + 𝐵)))
35 oveq2 6813 . . . . . . . . 9 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 6812 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 6812 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq1d 6820 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝑥𝑘) 𝐴) = ((𝑁𝑘) 𝐴))
3938oveq1d 6820 . . . . . . . . . 10 (𝑥 = 𝑁 → (((𝑥𝑘) 𝐴) × (𝑘 𝐵)) = (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))
4036, 39oveq12d 6823 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵))) = ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))
4135, 40mpteq12dv 4877 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥𝑘) 𝐴) × (𝑘 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵)))))
4241oveq2d 6821 . . . . . . 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 2767 . . . . . 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 1231 . . . . . . . . . . 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 18687 . . . . . . . . . . 11 𝑆 = (Base‘𝐺)
4945, 48syl6eleq 2841 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50 eqid 2752 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
51 eqid 2752 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
52 srgbinom.e . . . . . . . . . . 11 = (.g𝐺)
5350, 51, 52mulg0 17739 . . . . . . . . . 10 (𝐴 ∈ (Base‘𝐺) → (0 𝐴) = (0g𝐺))
5449, 53syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐴) = (0g𝐺))
55 simpr2 1233 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵𝑆)
5655, 48syl6eleq 2841 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
5750, 51, 52mulg0 17739 . . . . . . . . . 10 (𝐵 ∈ (Base‘𝐺) → (0 𝐵) = (0g𝐺))
5856, 57syl 17 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 𝐵) = (0g𝐺))
5954, 58oveq12d 6823 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 𝐴) × (0 𝐵)) = ((0g𝐺) × (0g𝐺)))
6059oveq2d 6821 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (1 · ((0g𝐺) × (0g𝐺))))
61 eqid 2752 . . . . . . . . . . . . . 14 (1r𝑅) = (1r𝑅)
6247, 61srgidcl 18710 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → (1r𝑅) ∈ 𝑆)
6362ancli 575 . . . . . . . . . . . 12 (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
6463adantr 472 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆))
65 srgbinom.m . . . . . . . . . . . 12 × = (.r𝑅)
6647, 65, 61srglidm 18713 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ (1r𝑅) ∈ 𝑆) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6764, 66syl 17 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r𝑅) × (1r𝑅)) = (1r𝑅))
6867oveq2d 6821 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · (1r𝑅)))
69 eqid 2752 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
7069, 61srgidcl 18710 . . . . . . . . . . 11 (𝑅 ∈ SRing → (1r𝑅) ∈ (Base‘𝑅))
71 srgbinom.t . . . . . . . . . . . 12 · = (.g𝑅)
7269, 71mulg1 17741 . . . . . . . . . . 11 ((1r𝑅) ∈ (Base‘𝑅) → (1 · (1r𝑅)) = (1r𝑅))
7370, 72syl 17 . . . . . . . . . 10 (𝑅 ∈ SRing → (1 · (1r𝑅)) = (1r𝑅))
7473adantr 472 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r𝑅)) = (1r𝑅))
7568, 74eqtrd 2786 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r𝑅) × (1r𝑅))) = (1r𝑅))
7646, 61ringidval 18695 . . . . . . . . 9 (1r𝑅) = (0g𝐺)
77 id 22 . . . . . . . . . . . 12 ((1r𝑅) = (0g𝐺) → (1r𝑅) = (0g𝐺))
7877, 77oveq12d 6823 . . . . . . . . . . 11 ((1r𝑅) = (0g𝐺) → ((1r𝑅) × (1r𝑅)) = ((0g𝐺) × (0g𝐺)))
7978oveq2d 6821 . . . . . . . . . 10 ((1r𝑅) = (0g𝐺) → (1 · ((1r𝑅) × (1r𝑅))) = (1 · ((0g𝐺) × (0g𝐺))))
8079, 77eqeq12d 2767 . . . . . . . . 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 208 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g𝐺) × (0g𝐺))) = (0g𝐺))
8360, 82eqtrd 2786 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) = (0g𝐺))
84 0z 11572 . . . . . . . . . . 11 0 ∈ ℤ
85 fzsn 12568 . . . . . . . . . . 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 4882 . . . . . . . 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 6821 . . . . . . 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 18701 . . . . . . . . 9 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
9190adantr 472 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
92 c0ex 10218 . . . . . . . . 9 0 ∈ V
9392a1i 11 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
9476, 62syl5eqelr 2836 . . . . . . . . . 10 (𝑅 ∈ SRing → (0g𝐺) ∈ 𝑆)
9594adantr 472 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g𝐺) ∈ 𝑆)
9683, 95eqeltrd 2831 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 𝐴) × (0 𝐵))) ∈ 𝑆)
97 oveq2 6813 . . . . . . . . . . 11 (𝑘 = 0 → (0C𝑘) = (0C0))
98 0nn0 11491 . . . . . . . . . . . 12 0 ∈ ℕ0
99 bcn0 13283 . . . . . . . . . . . 12 (0 ∈ ℕ0 → (0C0) = 1)
10098, 99ax-mp 5 . . . . . . . . . . 11 (0C0) = 1
10197, 100syl6eq 2802 . . . . . . . . . 10 (𝑘 = 0 → (0C𝑘) = 1)
102 oveq2 6813 . . . . . . . . . . . . 13 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
103 0m0e0 11314 . . . . . . . . . . . . 13 (0 − 0) = 0
104102, 103syl6eq 2802 . . . . . . . . . . . 12 (𝑘 = 0 → (0 − 𝑘) = 0)
105104oveq1d 6820 . . . . . . . . . . 11 (𝑘 = 0 → ((0 − 𝑘) 𝐴) = (0 𝐴))
106 oveq1 6812 . . . . . . . . . . 11 (𝑘 = 0 → (𝑘 𝐵) = (0 𝐵))
107105, 106oveq12d 6823 . . . . . . . . . 10 (𝑘 = 0 → (((0 − 𝑘) 𝐴) × (𝑘 𝐵)) = ((0 𝐴) × (0 𝐵)))
108101, 107oveq12d 6823 . . . . . . . . 9 (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))) = (1 · ((0 𝐴) × (0 𝐵))))
10947, 108gsumsn 18546 . . . . . . . 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 1473 . . . . . . 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 2786 . . . . . 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 17494 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑆𝐵𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
11491, 45, 55, 113syl3anc 1473 . . . . . . . 8 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
115114, 48syl6eleq 2841 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
11650, 51, 52mulg0 17739 . . . . . . 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 2797 . . . . 5 ((𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) 𝐴) × (𝑘 𝐵))))))
119 simprl 811 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑅 ∈ SRing)
12045adantl 473 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴𝑆)
12155adantl 473 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵𝑆)
122 simprr3 1274 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
123 simpl 474 . . . . . . . 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 18736 . . . . . . 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 631 . . . . . 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 11656 . . . 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  {csn 4313  cmpt 4873  cfv 6041  (class class class)co 6805  0cc0 10120  1c1 10121   + caddc 10123  cmin 10450  0cn0 11476  cz 11561  ...cfz 12511  Ccbc 13275  Basecbs 16051  +gcplusg 16135  .rcmulr 16136  0gc0g 16294   Σg cgsu 16295  Mndcmnd 17487  .gcmg 17733  mulGrpcmgp 18681  1rcur 18693  SRingcsrg 18697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-supp 7456  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8433  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-n0 11477  df-z 11562  df-uz 11872  df-rp 12018  df-fz 12512  df-fzo 12652  df-seq 12988  df-fac 13247  df-bc 13276  df-hash 13304  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-gsum 16297  df-mre 16440  df-mrc 16441  df-acs 16443  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-mhm 17528  df-submnd 17529  df-mulg 17734  df-cntz 17942  df-cmn 18387  df-mgp 18682  df-ur 18694  df-srg 18698
This theorem is referenced by:  csrgbinom  18738
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