Theorem srgbinom 20450

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 15992). (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 7425	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝐴 + 𝐵)) = (0 ↑ (𝐴 + 𝐵)))
2		oveq2 7426	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7425	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 − 𝑘) ↑ 𝐴) = ((0 − 𝑘) ↑ 𝐴))
6	5	oveq1d 7433	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
7	3, 6	oveq12d 7436	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
8	2, 7	mpteq12dv 5192	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
9	8	oveq2d 7434	. . . . . . 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 2777	. . . . . 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 343	. . . . 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 7425	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑛 ↑ (𝐴 + 𝐵)))
13		oveq2 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7425	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → ((𝑥 − 𝑘) ↑ 𝐴) = ((𝑛 − 𝑘) ↑ 𝐴))
17	16	oveq1d 7433	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
18	14, 17	oveq12d 7436	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
19	13, 18	mpteq12dv 5192	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
20	19	oveq2d 7434	. . . . . . 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 2777	. . . . . 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 343	. . . . 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 7425	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝐴 + 𝐵)) = ((𝑛 + 1) ↑ (𝐴 + 𝐵)))
24		oveq2 7426	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7425	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 − 𝑘) ↑ 𝐴) = (((𝑛 + 1) − 𝑘) ↑ 𝐴))
28	27	oveq1d 7433	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
29	25, 28	oveq12d 7436	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
30	24, 29	mpteq12dv 5192	. . . . . . . 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 7434	. . . . . . 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 2777	. . . . . 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 343	. . . . 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 7425	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑁 ↑ (𝐴 + 𝐵)))
35		oveq2 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7425	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → ((𝑥 − 𝑘) ↑ 𝐴) = ((𝑁 − 𝑘) ↑ 𝐴))
39	38	oveq1d 7433	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
40	36, 39	oveq12d 7436	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
41	35, 40	mpteq12dv 5192	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
42	41	oveq2d 7434	. . . . . . 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 2777	. . . . . 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 343	. . . . 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 1213	. . . . . . . . . . 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 20358	. . . . . . . . . . 11 ⊢ 𝑆 = (Base‘𝐺)
49	45, 48	eleqtrdi 2871	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
51		eqid 2761	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
52		srgbinom.e	. . . . . . . . . . 11 ⊢ ↑ = (.g‘𝐺)
53	50, 51, 52	mulg0 19277	. . . . . . . . . 10 ⊢ (𝐴 ∈ (Base‘𝐺) → (0 ↑ 𝐴) = (0g‘𝐺))
54	49, 53	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ 𝐴) = (0g‘𝐺))
55		simpr2 1214	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ 𝑆)
56	55, 48	eleqtrdi 2871	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
57	50, 51, 52	mulg0 19277	. . . . . . . . . 10 ⊢ (𝐵 ∈ (Base‘𝐺) → (0 ↑ 𝐵) = (0g‘𝐺))
58	56, 57	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ 𝐵) = (0g‘𝐺))
59	54, 58	oveq12d 7436	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 ↑ 𝐴) × (0 ↑ 𝐵)) = ((0g‘𝐺) × (0g‘𝐺)))
60	59	oveq2d 7434	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) = (1 · ((0g‘𝐺) × (0g‘𝐺))))
61		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑅) = (1r‘𝑅)
62	47, 61	srgidcl 20418	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ SRing → (1r‘𝑅) ∈ 𝑆)
63	62	ancli 558	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ SRing → (𝑅 ∈ SRing ∧ (1r‘𝑅) ∈ 𝑆))
64	63	adantr 486	. . . . . . . . . . 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 20421	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ (1r‘𝑅) ∈ 𝑆) → ((1r‘𝑅) × (1r‘𝑅)) = (1r‘𝑅))
67	64, 66	syl 18	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((1r‘𝑅) × (1r‘𝑅)) = (1r‘𝑅))
68	67	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1 · (1r‘𝑅)))
69		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
70	69, 61	srgidcl 20418	. . . . . . . . . . 11 ⊢ (𝑅 ∈ SRing → (1r‘𝑅) ∈ (Base‘𝑅))
71		srgbinom.t	. . . . . . . . . . . 12 ⊢ · = (.g‘𝑅)
72	69, 71	mulg1 19284	. . . . . . . . . . 11 ⊢ ((1r‘𝑅) ∈ (Base‘𝑅) → (1 · (1r‘𝑅)) = (1r‘𝑅))
73	70, 72	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (1 · (1r‘𝑅)) = (1r‘𝑅))
74	73	adantr 486	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · (1r‘𝑅)) = (1r‘𝑅))
75	68, 74	eqtrd 2796	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1r‘𝑅))
76	46, 61	ringidval 20402	. . . . . . . . 9 ⊢ (1r‘𝑅) = (0g‘𝐺)
77		id 23	. . . . . . . . . . . 12 ⊢ ((1r‘𝑅) = (0g‘𝐺) → (1r‘𝑅) = (0g‘𝐺))
78	77, 77	oveq12d 7436	. . . . . . . . . . 11 ⊢ ((1r‘𝑅) = (0g‘𝐺) → ((1r‘𝑅) × (1r‘𝑅)) = ((0g‘𝐺) × (0g‘𝐺)))
79	78	oveq2d 7434	. . . . . . . . . 10 ⊢ ((1r‘𝑅) = (0g‘𝐺) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1 · ((0g‘𝐺) × (0g‘𝐺))))
80	79, 77	eqeq12d 2777	. . . . . . . . 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 221	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0g‘𝐺) × (0g‘𝐺))) = (0g‘𝐺))
83	60, 82	eqtrd 2796	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) = (0g‘𝐺))
84		fz0sn 13754	. . . . . . . . . 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 5195	. . . . . . . 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 7434	. . . . . . 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 20409	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
89	88	adantr 486	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
90		c0ex 11293	. . . . . . . . 9 ⊢ 0 ∈ V
91	90	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
92	76, 62	eqeltrrid 2866	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (0g‘𝐺) ∈ 𝑆)
93	92	adantr 486	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0g‘𝐺) ∈ 𝑆)
94	83, 93	eqeltrd 2861	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) ∈ 𝑆)
95		oveq2 7426	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
96		0nn0 12614	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
97		bcn0 14447	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
98	96, 97	ax-mp 5	. . . . . . . . . . 11 ⊢ (0C0) = 1
99	95, 98	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
100		oveq2 7426	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
101		0m0e0 12454	. . . . . . . . . . . . 13 ⊢ (0 − 0) = 0
102	100, 101	eqtrdi 2812	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
103	102	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((0 − 𝑘) ↑ 𝐴) = (0 ↑ 𝐴))
104		oveq1 7425	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑘 ↑ 𝐵) = (0 ↑ 𝐵))
105	103, 104	oveq12d 7436	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = ((0 ↑ 𝐴) × (0 ↑ 𝐵)))
106	99, 105	oveq12d 7436	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))))
107	47, 106	gsumsn 20161	. . . . . . . 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 1398	. . . . . . 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 2796	. . . . . 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 18924	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
112	89, 45, 55, 111	syl3anc 1398	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
113	112, 48	eleqtrdi 2871	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
114	50, 51, 52	mulg0 19277	. . . . . . 7 ⊢ ((𝐴 + 𝐵) ∈ (Base‘𝐺) → (0 ↑ (𝐴 + 𝐵)) = (0g‘𝐺))
115	113, 114	syl 18	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (0 ↑ (𝐴 + 𝐵)) = (0g‘𝐺))
116	83, 109, 115	3eqtr4rd 2807	. . . . 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 783	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑅 ∈ SRing)
118	45	adantl 487	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐴 ∈ 𝑆)
119	55	adantl 487	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝐵 ∈ 𝑆)
120		simprr3 1242	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → (𝐴 × 𝐵) = (𝐵 × 𝐴))
121		simpl 488	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴)))) → 𝑛 ∈ ℕ0)
122		id 23	. . . . . . . 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 20449	. . . . . . 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 425	. . . . . 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 30	. . . . 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 12787	. . . 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 421	. . 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 413	. 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 412	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584   ↦ cmpt 5186  ‘cfv 6537  (class class class)co 7418  0cc0 11193  1c1 11194   + caddc 11196   − cmin 11534  ℕ₀cn0 12599  ...cfz 13632  Ccbc 14439  Basecbs 17380  +_gcplusg 17421  ._rcmulr 17422  0_gc0g 17603   Σ_g cgsu 17604  Mndcmnd 18916  ._gcmg 19270  mulGrpcmgp 20353  1_rcur 20400  SRingcsrg 20405

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-n0 12600  df-z 12687  df-uz 12959  df-rp 13114  df-fz 13633  df-fzo 13782  df-seq 14138  df-fac 14411  df-bc 14440  df-hash 14468  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-0g 17605  df-gsum 17606  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-submnd 18972  df-mulg 19271  df-cntz 19524  df-cmn 19989  df-mgp 20354  df-ur 20401  df-srg 20406

This theorem is used by:  csrgbinom  20451