Theorem srgbinom 20508

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 16043). (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 7458	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝐴 + 𝐵)) = (0 ↑ (𝐴 + 𝐵)))
2		oveq2 7459	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7458	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 − 𝑘) ↑ 𝐴) = ((0 − 𝑘) ↑ 𝐴))
6	5	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
7	3, 6	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
8	2, 7	mpteq12dv 5198	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...0) ↦ ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
9	8	oveq2d 7467	. . . . . . 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 2779	. . . . . 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 7458	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑛 ↑ (𝐴 + 𝐵)))
13		oveq2 7459	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7458	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → ((𝑥 − 𝑘) ↑ 𝐴) = ((𝑛 − 𝑘) ↑ 𝐴))
17	16	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
18	14, 17	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
19	13, 18	mpteq12dv 5198	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑛C𝑘) · (((𝑛 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
20	19	oveq2d 7467	. . . . . . 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 2779	. . . . . 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 7458	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝐴 + 𝐵)) = ((𝑛 + 1) ↑ (𝐴 + 𝐵)))
24		oveq2 7459	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7458	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 − 𝑘) ↑ 𝐴) = (((𝑛 + 1) − 𝑘) ↑ 𝐴))
28	27	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
29	25, 28	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = (((𝑛 + 1)C𝑘) · ((((𝑛 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
30	24, 29	mpteq12dv 5198	. . . . . . . 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 7467	. . . . . . 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 2779	. . . . . 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 7458	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝐴 + 𝐵)) = (𝑁 ↑ (𝐴 + 𝐵)))
35		oveq2 7459	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7458	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → ((𝑥 − 𝑘) ↑ 𝐴) = ((𝑁 − 𝑘) ↑ 𝐴))
39	38	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))
40	36, 39	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))
41	35, 40	mpteq12dv 5198	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ (0...𝑥) ↦ ((𝑥C𝑘) · (((𝑥 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))
42	41	oveq2d 7467	. . . . . . 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 2779	. . . . . 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 20416	. . . . . . . . . . 11 ⊢ 𝑆 = (Base‘𝐺)
49	45, 48	eleqtrdi 2874	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐴 ∈ (Base‘𝐺))
50		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
51		eqid 2763	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
52		srgbinom.e	. . . . . . . . . . 11 ⊢ ↑ = (.g‘𝐺)
53	50, 51, 52	mulg0 19334	. . . . . . . . . 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 2874	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝐵 ∈ (Base‘𝐺))
57	50, 51, 52	mulg0 19334	. . . . . . . . . 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 7469	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → ((0 ↑ 𝐴) × (0 ↑ 𝐵)) = ((0g‘𝐺) × (0g‘𝐺)))
60	59	oveq2d 7467	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) = (1 · ((0g‘𝐺) × (0g‘𝐺))))
61		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑅) = (1r‘𝑅)
62	47, 61	srgidcl 20476	. . . . . . . . . . . . 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 20479	. . . . . . . . . . 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 7467	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1 · (1r‘𝑅)))
69		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
70	69, 61	srgidcl 20476	. . . . . . . . . . 11 ⊢ (𝑅 ∈ SRing → (1r‘𝑅) ∈ (Base‘𝑅))
71		srgbinom.t	. . . . . . . . . . . 12 ⊢ · = (.g‘𝑅)
72	69, 71	mulg1 19341	. . . . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1r‘𝑅))
76	46, 61	ringidval 20460	. . . . . . . . 9 ⊢ (1r‘𝑅) = (0g‘𝐺)
77		id 23	. . . . . . . . . . . 12 ⊢ ((1r‘𝑅) = (0g‘𝐺) → (1r‘𝑅) = (0g‘𝐺))
78	77, 77	oveq12d 7469	. . . . . . . . . . 11 ⊢ ((1r‘𝑅) = (0g‘𝐺) → ((1r‘𝑅) × (1r‘𝑅)) = ((0g‘𝐺) × (0g‘𝐺)))
79	78	oveq2d 7467	. . . . . . . . . 10 ⊢ ((1r‘𝑅) = (0g‘𝐺) → (1 · ((1r‘𝑅) × (1r‘𝑅))) = (1 · ((0g‘𝐺) × (0g‘𝐺))))
80	79, 77	eqeq12d 2779	. . . . . . . . 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 2798	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) = (0g‘𝐺))
84		fz0sn 13801	. . . . . . . . . 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 5201	. . . . . . . 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 7467	. . . . . . 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 20467	. . . . . . . . 9 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
89	88	adantr 486	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 𝑅 ∈ Mnd)
90		c0ex 11339	. . . . . . . . 9 ⊢ 0 ∈ V
91	90	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → 0 ∈ V)
92	76, 62	eqeltrrid 2869	. . . . . . . . . 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 2864	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))) ∈ 𝑆)
95		oveq2 7459	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
96		0nn0 12661	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ0
97		bcn0 14495	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
98	96, 97	ax-mp 5	. . . . . . . . . . 11 ⊢ (0C0) = 1
99	95, 98	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
100		oveq2 7459	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
101		0m0e0 12501	. . . . . . . . . . . . 13 ⊢ (0 − 0) = 0
102	100, 101	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
103	102	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((0 − 𝑘) ↑ 𝐴) = (0 ↑ 𝐴))
104		oveq1 7458	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑘 ↑ 𝐵) = (0 ↑ 𝐵))
105	103, 104	oveq12d 7469	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)) = ((0 ↑ 𝐴) × (0 ↑ 𝐵)))
106	99, 105	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((0C𝑘) · (((0 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))) = (1 · ((0 ↑ 𝐴) × (0 ↑ 𝐵))))
107	47, 106	gsumsn 20219	. . . . . . . 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 1399	. . . . . . 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 2798	. . . . . 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 18981	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 + 𝐵) ∈ 𝑆)
112	89, 45, 55, 111	syl3anc 1399	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ 𝑆)
113	112, 48	eleqtrdi 2874	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
114	50, 51, 52	mulg0 19334	. . . . . . 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 2809	. . . . 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 20507	. . . . . . 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 12834	. . . 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 2147  Vcvv 3454  {csn 4588   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451  0cc0 11239  1c1 11240   + caddc 11242   − cmin 11580  ℕ₀cn0 12646  ...cfz 13679  Ccbc 14487  Basecbs 17431  +_gcplusg 17472  ._rcmulr 17473  0_gc0g 17660   Σ_g cgsu 17661  Mndcmnd 18973  ._gcmg 19327  mulGrpcmgp 20411  1_rcur 20458  SRingcsrg 20463

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-seq 14185  df-fac 14459  df-bc 14488  df-hash 14516  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-0g 17662  df-gsum 17663  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-mulg 19328  df-cntz 19582  df-cmn 20047  df-mgp 20412  df-ur 20459  df-srg 20464

This theorem is used by:  csrgbinom  20509