Theorem srgbinom 20144

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 15732). (Contributed by AV, 24-Aug-2019.)

Hypotheses

Ref	Expression
srgbinom.s	⊢ 𝑆 = (Base‘𝑅)
srgbinom.m	⊢ × = (.r‘𝑅)
srgbinom.t	⊢ · = (.g‘𝑅)
srgbinom.a	⊢ + = (+g‘𝑅)
srgbinom.g	⊢ 𝐺 = (mulGrp‘𝑅)
srgbinom.e	⊢ ↑ = (.g‘𝐺)

Assertion

Ref	Expression
srgbinom	⊢ (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 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.

Step	Hyp	Ref	Expression
1		oveq1 7348	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝐴 + 𝐵)) = (0 ↑ (𝐴 + 𝐵)))
2		oveq2 7349	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7348	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 − 𝑘) ↑ 𝐴) = ((0 − 𝑘) ↑ 𝐴))
6	5	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
7	3, 6	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
8	2, 7	mpteq12dv 5173	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
9	8	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
10	1, 9	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) ↔ (0 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))))
11	10	imbi2d 340	. . . . 5 ⊢ (𝑥 = 0 → (((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))))
12		oveq1 7348	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑛 ↑ (𝐴 + 𝐵)))
13		oveq2 7349	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7348	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → ((𝑥 − 𝑘) ↑ 𝐴) = ((𝑛 − 𝑘) ↑ 𝐴))
17	16	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
18	14, 17	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
19	13, 18	mpteq12dv 5173	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
20	19	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
21	12, 20	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) ↔ (𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))))
22	21	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑛 → (((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))))
23		oveq1 7348	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝐴 + 𝐵)) = ((𝑛 + 1) ↑ (𝐴 + 𝐵)))
24		oveq2 7349	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7348	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 − 𝑘) ↑ 𝐴) = (((𝑛 + 1) − 𝑘) ↑ 𝐴))
28	27	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
29	25, 28	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
30	24, 29	mpteq12dv 5173	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
31	30	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
32	23, 31	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) ↔ ((𝑛 + 1) ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))))
33	32	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (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 7348	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑁 ↑ (𝐴 + 𝐵)))
35		oveq2 7349	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7348	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → ((𝑥 − 𝑘) ↑ 𝐴) = ((𝑁 − 𝑘) ↑ 𝐴))
39	38	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
40	36, 39	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
41	35, 40	mpteq12dv 5173	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
42	41	oveq2d 7357	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
43	34, 42	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) ↔ (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))))
44	43	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑁 → (((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ↔ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))))
45		simpr1 1195	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ 𝑆)
46		srgbinom.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (mulGrp‘𝑅)
47		srgbinom.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (Base‘𝑅)
48	46, 47	mgpbas 20058	. . . . . . . . . . 11 ⊢ 𝑆 = (Base‘𝐺)
49	45, 48	eleqtrdi 2841	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
51		eqid 2731	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
52		srgbinom.e	. . . . . . . . . . 11 ⊢ ↑ = (.g‘𝐺)
53	50, 51, 52	mulg0 18982	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Base‘𝐺) → (0 ↑ 𝐴) = (0g‘𝐺))
54	49, 53	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ 𝐴) = (0g‘𝐺))
55		simpr2 1196	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ 𝑆)
56	55, 48	eleqtrdi 2841	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
57	50, 51, 52	mulg0 18982	. . . . . . . . . 10 ⊢ (𝐵 ∈ (Base‘𝐺) → (0 ↑ 𝐵) = (0g‘𝐺))
58	56, 57	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ 𝐵) = (0g‘𝐺))
59	54, 58	oveq12d 7359	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 ↑ 𝐴) × (0 ↑ 𝐵)) = ((0g‘𝐺) × (0g‘𝐺)))
60	59	oveq2d 7357	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) = (1 · ((0g‘𝐺) × (0g‘𝐺))))
61		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑅) = (1r‘𝑅)
62	47, 61	srgidcl 20112	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ SRing → (1r‘𝑅) ∈ 𝑆)
63	62	ancli 548	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r‘𝑅) ∈ 𝑆))
64	63	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 ∈ SRing ∧ (1r‘𝑅) ∈ 𝑆))
65		srgbinom.m	. . . . . . . . . . . 12 ⊢ × = (.r‘𝑅)
66	47, 65, 61	srglidm 20115	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ (1r‘𝑅) ∈ 𝑆) → ((1r‘𝑅) × (1r‘𝑅)) = (1r‘𝑅))
67	64, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r‘𝑅) × (1r‘𝑅)) = (1r‘𝑅))
68	67	oveq2d 7357	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1 · (1r‘𝑅)))
69		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
70	69, 61	srgidcl 20112	. . . . . . . . . . 11 ⊢ (𝑅 ∈ SRing → (1r‘𝑅) ∈ (Base‘𝑅))
71		srgbinom.t	. . . . . . . . . . . 12 ⊢ · = (.g‘𝑅)
72	69, 71	mulg1 18989	. . . . . . . . . . 11 ⊢ ((1r‘𝑅) ∈ (Base‘𝑅) → (1 · (1r‘𝑅)) = (1r‘𝑅))
73	70, 72	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (1 · (1r‘𝑅)) = (1r‘𝑅))
74	73	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r‘𝑅)) = (1r‘𝑅))
75	68, 74	eqtrd 2766	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1r‘𝑅))
76	46, 61	ringidval 20096	. . . . . . . . 9 ⊢ (1r‘𝑅) = (0g‘𝐺)
77		id 22	. . . . . . . . . . . 12 ⊢ ((1r‘𝑅) = (0g‘𝐺) → (1r‘𝑅) = (0g‘𝐺))
78	77, 77	oveq12d 7359	. . . . . . . . . . 11 ⊢ ((1r‘𝑅) = (0g‘𝐺) → ((1r‘𝑅) × (1r‘𝑅)) = ((0g‘𝐺) × (0g‘𝐺)))
79	78	oveq2d 7357	. . . . . . . . . 10 ⊢ ((1r‘𝑅) = (0g‘𝐺) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1 · ((0g‘𝐺) × (0g‘𝐺))))
80	79, 77	eqeq12d 2747	. . . . . . . . 9 ⊢ ((1r‘𝑅) = (0g‘𝐺) → ((1 · ((1r‘𝑅) × (1r‘𝑅))) = (1r‘𝑅) ↔ (1 · ((0g‘𝐺) × (0g‘𝐺))) = (0g‘𝐺)))
81	76, 80	ax-mp 5	. . . . . . . 8 ⊢ ((1 · ((1r‘𝑅) × (1r‘𝑅))) = (1r‘𝑅) ↔ (1 · ((0g‘𝐺) × (0g‘𝐺))) = (0g‘𝐺))
82	75, 81	sylib 218	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g‘𝐺) × (0g‘𝐺))) = (0g‘𝐺))
83	60, 82	eqtrd 2766	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) = (0g‘𝐺))
84		fz0sn 13522	. . . . . . . . . 10 ⊢ (0...0) = {0}
85	84	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0...0) = {0})
86	85	mpteq1d 5176	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
87	86	oveq2d 7357	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (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 20103	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
89	88	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
90		c0ex 11101	. . . . . . . . 9 ⊢ 0 ∈ V
91	90	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
92	76, 62	eqeltrrid 2836	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (0g‘𝐺) ∈ 𝑆)
93	92	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g‘𝐺) ∈ 𝑆)
94	83, 93	eqeltrd 2831	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) ∈ 𝑆)
95		oveq2 7349	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
96		0nn0 12391	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
97		bcn0 14212	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
98	96, 97	ax-mp 5	. . . . . . . . . . 11 ⊢ (0C0) = 1
99	95, 98	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
100		oveq2 7349	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
101		0m0e0 12235	. . . . . . . . . . . . 13 ⊢ (0 − 0) = 0
102	100, 101	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
103	102	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((0 − 𝑘) ↑ 𝐴) = (0 ↑ 𝐴))
104		oveq1 7348	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑘 ↑ 𝐵) = (0 ↑ 𝐵))
105	103, 104	oveq12d 7359	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = ((0 ↑ 𝐴) × (0 ↑ 𝐵)))
106	99, 105	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))))
107	47, 106	gsumsn 19861	. . . . . . . 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))))
108	89, 91, 94, 107	syl3anc 1373	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ {0} ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) = (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))))
109	87, 108	eqtrd 2766	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑅 Σg (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) = (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))))
110		srgbinom.a	. . . . . . . . . 10 ⊢ + = (+g‘𝑅)
111	47, 110	mndcl 18645	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
112	89, 45, 55, 111	syl3anc 1373	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
113	112, 48	eleqtrdi 2841	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
114	50, 51, 52	mulg0 18982	. . . . . . 7 ⊢ ((𝐴 + 𝐵) ∈ (Base‘𝐺) → (0 ↑ (𝐴 + 𝐵)) = (0g‘𝐺))
115	113, 114	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ (𝐴 + 𝐵)) = (0g‘𝐺))
116	83, 109, 115	3eqtr4rd 2777	. . . . 5 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (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 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑅 ∈ SRing)
118	45	adantl 481	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴 ∈ 𝑆)
119	55	adantl 481	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵 ∈ 𝑆)
120		simprr3 1224	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
121		simpl 482	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑛 ∈ ℕ0)
122		id 22	. . . . . . . 8 ⊢ ((𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) → (𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
123	47, 65, 71, 110, 46, 52, 117, 118, 119, 120, 121, 122	srgbinomlem 20143	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) ∧ (𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) → ((𝑛 + 1) ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
124	123	exp31 419	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))) → ((𝑛 + 1) ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))))
125	124	a2d 29	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑛 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) → ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((𝑛 + 1) ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑛 + 1)) ↦ (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))))
126	11, 22, 33, 44, 116, 125	nn0ind 12563	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))))
127	126	expd 415	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ SRing → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))))
128	127	impcom 407	. 2 ⊢ ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))))
129	128	imp 406	1 ⊢ (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4571   ↦ cmpt 5167  ‘cfv 6476  (class class class)co 7341  0cc0 11001  1c1 11002   + caddc 11004   − cmin 11339  ℕ₀cn0 12376  ...cfz 13402  Ccbc 14204  Basecbs 17115  +_gcplusg 17156  ._rcmulr 17157  0_gc0g 17338   Σ_g cgsu 17339  Mndcmnd 18637  ._gcmg 18975  mulGrpcmgp 20053  1_rcur 20094  SRingcsrg 20099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-oi 9391  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-n0 12377  df-z 12464  df-uz 12728  df-rp 12886  df-fz 13403  df-fzo 13550  df-seq 13904  df-fac 14176  df-bc 14205  df-hash 14233  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-0g 17340  df-gsum 17341  df-mre 17483  df-mrc 17484  df-acs 17486  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-mulg 18976  df-cntz 19224  df-cmn 19689  df-mgp 20054  df-ur 20095  df-srg 20100

This theorem is referenced by:  csrgbinom  20145