Theorem xpsmnd0 18737

Description: The identity element of a binary product of monoids. (Contributed by AV, 25-Feb-2025.)

Hypothesis

Ref	Expression
xpsmnd0.t	⊢ 𝑇 = (𝑅 ×s 𝑆)

Assertion

Ref	Expression
xpsmnd0	⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)

Proof of Theorem xpsmnd0

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
2		eqid 2737	. . 3 ⊢ (0g‘𝑇) = (0g‘𝑇)
3		eqid 2737	. . 3 ⊢ (+g‘𝑇) = (+g‘𝑇)
4		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
6	4, 5	mndidcl 18708	. . . . . 6 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅))
7	6	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑅) ∈ (Base‘𝑅))
8		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	8, 9	mndidcl 18708	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ (Base‘𝑆))
11	10	adantl 481	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑆) ∈ (Base‘𝑆))
12	7, 11	opelxpd 5663	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ ∈ ((Base‘𝑅) × (Base‘𝑆)))
13		xpsmnd0.t	. . . . 5 ⊢ 𝑇 = (𝑅 ×s 𝑆)
14		simpl 482	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑅 ∈ Mnd)
15		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑆 ∈ Mnd)
16	13, 4, 8, 14, 15	xpsbas 17527	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ((Base‘𝑅) × (Base‘𝑆)) = (Base‘𝑇))
17	12, 16	eleqtrd 2839	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ ∈ (Base‘𝑇))
18	16	eleq2d 2823	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ 𝑥 ∈ (Base‘𝑇)))
19		elxp2 5648	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩)
20	14	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑅 ∈ Mnd)
21	15	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑆 ∈ Mnd)
22	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (0g‘𝑅) ∈ (Base‘𝑅))
23	11	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (0g‘𝑆) ∈ (Base‘𝑆))
24		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆)) → 𝑎 ∈ (Base‘𝑅))
25	24	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑎 ∈ (Base‘𝑅))
26		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆)) → 𝑏 ∈ (Base‘𝑆))
27	26	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑏 ∈ (Base‘𝑆))
28		eqid 2737	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
29	4, 28	mndcl 18701	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
30	20, 22, 25, 29	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
31		eqid 2737	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	8, 31	mndcl 18701	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ (0g‘𝑆) ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	21, 23, 27, 32	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑆)(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
34	13, 4, 8, 20, 21, 22, 23, 25, 27, 30, 33, 28, 31, 3	xpsadd 17529	. . . . . . . . 9 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩) = ⟨((0g‘𝑅)(+g‘𝑅)𝑎), ((0g‘𝑆)(+g‘𝑆)𝑏)⟩)
35	4, 28, 5	mndlid 18713	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) = 𝑎)
36	14, 24, 35	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) = 𝑎)
37	8, 31, 9	mndlid 18713	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) = 𝑏)
38	15, 26, 37	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑆)(+g‘𝑆)𝑏) = 𝑏)
39	36, 38	opeq12d 4825	. . . . . . . . 9 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ⟨((0g‘𝑅)(+g‘𝑅)𝑎), ((0g‘𝑆)(+g‘𝑆)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
40	34, 39	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
41		oveq2 7368	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩))
42		id 22	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → 𝑥 = ⟨𝑎, 𝑏⟩)
43	41, 42	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥 ↔ (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩))
44	40, 43	syl5ibrcom 247	. . . . . . 7 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥))
45	44	rexlimdvva 3195	. . . . . 6 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥))
46	19, 45	biimtrid 242	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥))
47	18, 46	sylbird 260	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ (Base‘𝑇) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥))
48	47	imp 406	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝑇)) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥)
49	4, 28	mndcl 18701	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅) ∧ (0g‘𝑅) ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
50	20, 25, 22, 49	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
51	8, 31	mndcl 18701	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) ∈ (Base‘𝑆))
52	21, 27, 23, 51	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑏(+g‘𝑆)(0g‘𝑆)) ∈ (Base‘𝑆))
53	13, 4, 8, 20, 21, 25, 27, 22, 23, 50, 52, 28, 31, 3	xpsadd 17529	. . . . . . . . 9 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = ⟨(𝑎(+g‘𝑅)(0g‘𝑅)), (𝑏(+g‘𝑆)(0g‘𝑆))⟩)
54	4, 28, 5	mndrid 18714	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) = 𝑎)
55	14, 24, 54	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) = 𝑎)
56	8, 31, 9	mndrid 18714	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) = 𝑏)
57	15, 26, 56	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑏(+g‘𝑆)(0g‘𝑆)) = 𝑏)
58	55, 57	opeq12d 4825	. . . . . . . . 9 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ⟨(𝑎(+g‘𝑅)(0g‘𝑅)), (𝑏(+g‘𝑆)(0g‘𝑆))⟩ = ⟨𝑎, 𝑏⟩)
59	53, 58	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7367	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩))
61	60, 42	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥 ↔ (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = ⟨𝑎, 𝑏⟩))
62	59, 61	syl5ibrcom 247	. . . . . . 7 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥))
63	62	rexlimdvva 3195	. . . . . 6 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥))
64	19, 63	biimtrid 242	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥))
65	18, 64	sylbird 260	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ (Base‘𝑇) → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥))
66	65	imp 406	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝑇)) → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥)
67	1, 2, 3, 17, 48, 66	ismgmid2 18627	. 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ = (0g‘𝑇))
68	67	eqcomd 2743	1 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ⟨cop 4574   × cxp 5622  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393   ×_s cxps 17461  Mndcmnd 18693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-prds 17401  df-imas 17463  df-xps 17465  df-mgm 18599  df-sgrp 18678  df-mnd 18694

This theorem is referenced by:  xpsinv  19027  rngqiprngimf1  21290