Theorem srgmulgass 14267

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 6082	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
2	1	oveq1d 6090	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3		oveq1 6082	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
4	2, 3	eqeq12d 2253	. . . . . 6 ⊢ (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
5	4	imbi2d 230	. . . . 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 6082	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
7	6	oveq1d 6090	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8		oveq1 6082	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
9	7, 8	eqeq12d 2253	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
10	9	imbi2d 230	. . . . 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 6082	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
12	11	oveq1d 6090	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13		oveq1 6082	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
14	12, 13	eqeq12d 2253	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
15	14	imbi2d 230	. . . . 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 6082	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
17	16	oveq1d 6090	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18		oveq1 6082	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
19	17, 18	eqeq12d 2253	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
20	19	imbi2d 230	. . . . 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 110	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22		simpr 110	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
23	22	adantr 276	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑌 ∈ 𝐵)
24		srgmulgass.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
25		srgmulgass.t	. . . . . . . 8 ⊢ × = (.r‘𝑅)
26		eqid 2238	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	24, 25, 26	srglz 14263	. . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑌 ∈ 𝐵) → ((0g‘𝑅) × 𝑌) = (0g‘𝑅))
28	21, 23, 27	syl2anc 415	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0g‘𝑅) × 𝑌) = (0g‘𝑅))
29		simpl 109	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
30	29	adantr 276	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑋 ∈ 𝐵)
31		srgmulgass.m	. . . . . . . . 9 ⊢ · = (.g‘𝑅)
32	24, 26, 31	mulg0 13905	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝑅))
33	30, 32	syl 14	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g‘𝑅))
34	33	oveq1d 6090	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g‘𝑅) × 𝑌))
35	24, 25	srgcl 14248	. . . . . . . 8 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
36	21, 30, 23, 35	syl3anc 1278	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
37	24, 26, 31	mulg0 13905	. . . . . . 7 ⊢ ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g‘𝑅))
38	36, 37	syl 14	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g‘𝑅))
39	28, 34, 38	3eqtr4d 2281	. . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40		srgmnd 14245	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
41	40	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ Mnd)
42	41	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ Mnd)
43		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
44	30	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋 ∈ 𝐵)
45		eqid 2238	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
46	24, 31, 45	mulgnn0p1 13913	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)𝑋))
47	42, 43, 44, 46	syl3anc 1278	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝑅)𝑋))
48	47	oveq1d 6090	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌))
49	21	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ SRing)
50	24, 31	mulgnn0cl 13918	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
51	42, 43, 44, 50	syl3anc 1278	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
52	23	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌 ∈ 𝐵)
53	24, 45, 25	srgdir 14253	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
54	49, 51, 44, 52, 53	syl13anc 1280	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g‘𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
55	48, 54	eqtrd 2271	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)))
56	55	adantr 276	. . . . . . . 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)))
57		oveq1 6082	. . . . . . . . 9 ⊢ (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
58	35	3expb 1235	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
59	58	ancoms 268	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
60	59	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
61	24, 31, 45	mulgnn0p1 13913	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
62	42, 43, 60, 61	syl3anc 1278	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)))
63	62	eqcomd 2244	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
64	57, 63	sylan9eqr 2293	. . . . . . . 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)))
65	56, 64	eqtrd 2271	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
66	65	exp31 364	. . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
67	66	a2d 26	. . . . 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)))))
68	5, 10, 15, 20, 39, 67	nn0ind 9739	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
69	68	expd 258	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
70	69	3impib 1232	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
71	70	impcom 125	1 ⊢ ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ‘cfv 5372  (class class class)co 6075  0cc0 8169  1c1 8170   + caddc 8172  ℕ₀cn0 9542  Basecbs 13330  +_gcplusg 13408  ._rcmulr 13409  0_gc0g 13587  Mndcmnd 13706  ._gcmg 13899  SRingcsrg 14241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-minusg 13786  df-mulg 13900  df-cmn 14066  df-mgp 14195  df-srg 14242

This theorem is referenced by:  srgpcomppsc  14270