Theorem srgmulgass 20443

Description: An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)

Hypotheses

Ref	Expression
srgmulgass.b	⊢ 𝐵 = (Base‘𝑅)
srgmulgass.m	⊢ · = (.g‘𝑅)
srgmulgass.t	⊢ × = (.r‘𝑅)

Assertion

Ref	Expression
srgmulgass	⊢ ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Proof of Theorem srgmulgass

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
2	1	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3		oveq1 7427	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
4	2, 3	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
5	4	imbi2d 343	. . . . 5 ⊢ (𝑥 = 0 → ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))))
6		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
7	6	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8		oveq1 7427	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
9	7, 8	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
10	9	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑦 → ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)))))
11		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
12	11	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13		oveq1 7427	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
14	12, 13	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
15	14	imbi2d 343	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
16		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
17	16	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18		oveq1 7427	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
19	17, 18	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
20	19	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑁 → ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
21		simpr 490	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22		simpr 490	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
23	22	adantr 486	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑌 ∈ 𝐵)
24		srgmulgass.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
25		srgmulgass.t	. . . . . . . 8 ⊢ × = (.r‘𝑅)
26		eqid 2760	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	24, 25, 26	srglz 20434	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑌 ∈ 𝐵) → ((0g‘𝑅) × 𝑌) = (0g‘𝑅))
28	21, 23, 27	syl2anc 596	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0g‘𝑅) × 𝑌) = (0g‘𝑅))
29		simpl 488	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
30	29	adantr 486	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑋 ∈ 𝐵)
31		srgmulgass.m	. . . . . . . . 9 ⊢ · = (.g‘𝑅)
32	24, 26, 31	mulg0 19284	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝑅))
33	30, 32	syl 18	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g‘𝑅))
34	33	oveq1d 7435	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g‘𝑅) × 𝑌))
35	24, 25	srgcl 20419	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
36	21, 30, 23, 35	syl3anc 1398	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
37	24, 26, 31	mulg0 19284	. . . . . . 7 ⊢ ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g‘𝑅))
38	36, 37	syl 18	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g‘𝑅))
39	28, 34, 38	3eqtr4d 2805	. . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40		srgmnd 20416	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
41	40	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ Mnd)
42	41	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ Mnd)
43		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
44	30	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋 ∈ 𝐵)
45		eqid 2760	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
46	24, 31, 45	mulgnn0p1 19295	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)𝑋))
47	42, 43, 44, 46	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)𝑋))
48	47	oveq1d 7435	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌))
49	21	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ SRing)
50	24, 31, 42, 43, 44	mulgnn0cld 19305	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
51	23	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌 ∈ 𝐵)
52	24, 45, 25	srgdir 20424	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
53	49, 50, 44, 51, 52	syl13anc 1399	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
54	48, 53	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
55	54	adantr 486	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
56		oveq1 7427	. . . . . . . . 9 ⊢ (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
57	35	3expb 1138	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
58	57	ancoms 464	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
59	58	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
60	24, 31, 45	mulgnn0p1 19295	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
61	42, 43, 59, 60	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
62	61	eqcomd 2766	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
63	56, 62	sylan9eqr 2817	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
64	55, 63	eqtrd 2795	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
65	64	exp31 425	. . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
66	65	a2d 30	. . . . 5 ⊢ (𝑦 ∈ ℕ0 → ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
67	5, 10, 15, 20, 39, 66	nn0ind 12794	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
68	67	expd 421	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
69	68	3impib 1134	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
70	69	impcom 413	1 ⊢ ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420  0cc0 11200  1c1 11201   + caddc 11203  ℕ₀cn0 12606  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429  0_gc0g 17610  Mndcmnd 18923  ._gcmg 19277  SRingcsrg 20412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-seq 14145  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-plusg 17441  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mulg 19278  df-cmn 19996  df-mgp 20361  df-srg 20413

This theorem is used by:  srgpcomppsc  20446  srgbinomlem4  20455