Theorem srgmulgass 20244

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 7455	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
2	1	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3		oveq1 7455	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
4	2, 3	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
5	4	imbi2d 340	. . . . 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 7455	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
7	6	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8		oveq1 7455	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
9	7, 8	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
10	9	imbi2d 340	. . . . 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 7455	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
12	11	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13		oveq1 7455	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
14	12, 13	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
15	14	imbi2d 340	. . . . 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 7455	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
17	16	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18		oveq1 7455	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
19	17, 18	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
20	19	imbi2d 340	. . . . 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 484	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22		simpr 484	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
23	22	adantr 480	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑌 ∈ 𝐵)
24		srgmulgass.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
25		srgmulgass.t	. . . . . . . 8 ⊢ × = (.r‘𝑅)
26		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	24, 25, 26	srglz 20235	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑌 ∈ 𝐵) → ((0g‘𝑅) × 𝑌) = (0g‘𝑅))
28	21, 23, 27	syl2anc 583	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0g‘𝑅) × 𝑌) = (0g‘𝑅))
29		simpl 482	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑋 ∈ 𝐵)
31		srgmulgass.m	. . . . . . . . 9 ⊢ · = (.g‘𝑅)
32	24, 26, 31	mulg0 19114	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝑅))
33	30, 32	syl 17	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g‘𝑅))
34	33	oveq1d 7463	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g‘𝑅) × 𝑌))
35	24, 25	srgcl 20220	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
36	21, 30, 23, 35	syl3anc 1371	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
37	24, 26, 31	mulg0 19114	. . . . . . 7 ⊢ ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g‘𝑅))
38	36, 37	syl 17	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g‘𝑅))
39	28, 34, 38	3eqtr4d 2790	. . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40		srgmnd 20217	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
41	40	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ Mnd)
42	41	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ Mnd)
43		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
44	30	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋 ∈ 𝐵)
45		eqid 2740	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
46	24, 31, 45	mulgnn0p1 19125	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)𝑋))
47	42, 43, 44, 46	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)𝑋))
48	47	oveq1d 7463	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌))
49	21	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ SRing)
50	24, 31, 42, 43, 44	mulgnn0cld 19135	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
51	23	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌 ∈ 𝐵)
52	24, 45, 25	srgdir 20225	. . . . . . . . . . 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 1372	. . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
55	54	adantr 480	. . . . . . . 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 7455	. . . . . . . . 9 ⊢ (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
57	35	3expb 1120	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
58	57	ancoms 458	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
59	58	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
60	24, 31, 45	mulgnn0p1 19125	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
62	61	eqcomd 2746	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
63	56, 62	sylan9eqr 2802	. . . . . . . 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 2780	. . . . . . 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 419	. . . . . 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 29	. . . . 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 12738	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
68	67	expd 415	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
69	68	3impib 1116	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
70	69	impcom 407	1 ⊢ ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  0cc0 11184  1c1 11185   + caddc 11187  ℕ₀cn0 12553  Basecbs 17258  +_gcplusg 17311  ._rcmulr 17312  0_gc0g 17499  Mndcmnd 18772  ._gcmg 19107  SRingcsrg 20213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mulg 19108  df-cmn 19824  df-mgp 20162  df-srg 20214

This theorem is referenced by:  srgpcomppsc  20247  srgbinomlem4  20256