Theorem xpsmnd0 18804

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 2735	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
2		eqid 2735	. . 3 ⊢ (0g‘𝑇) = (0g‘𝑇)
3		eqid 2735	. . 3 ⊢ (+g‘𝑇) = (+g‘𝑇)
4		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2735	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
6	4, 5	mndidcl 18775	. . . . . 6 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅))
7	6	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑅) ∈ (Base‘𝑅))
8		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2735	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	8, 9	mndidcl 18775	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ (Base‘𝑆))
11	10	adantl 481	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑆) ∈ (Base‘𝑆))
12	7, 11	opelxpd 5728	. . . 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 17619	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ((Base‘𝑅) × (Base‘𝑆)) = (Base‘𝑇))
17	12, 16	eleqtrd 2841	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ ∈ (Base‘𝑇))
18	16	eleq2d 2825	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ 𝑥 ∈ (Base‘𝑇)))
19		elxp2 5713	. . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
29	4, 28	mndcl 18768	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
30	20, 22, 25, 29	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
31		eqid 2735	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	8, 31	mndcl 18768	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ (0g‘𝑆) ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	21, 23, 27, 32	syl3anc 1370	. . . . . . . . . 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 17621	. . . . . . . . 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 18780	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) = 𝑎)
36	14, 24, 35	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) = 𝑎)
37	8, 31, 9	mndlid 18780	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) = 𝑏)
38	15, 26, 37	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑆)(+g‘𝑆)𝑏) = 𝑏)
39	36, 38	opeq12d 4886	. . . . . . . . 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 2775	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
41		oveq2 7439	. . . . . . . . 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 2751	. . . . . . . 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 3211	. . . . . 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 18768	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅) ∧ (0g‘𝑅) ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
50	20, 25, 22, 49	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
51	8, 31	mndcl 18768	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) ∈ (Base‘𝑆))
52	21, 27, 23, 51	syl3anc 1370	. . . . . . . . . 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 17621	. . . . . . . . 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 18781	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) = 𝑎)
55	14, 24, 54	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) = 𝑎)
56	8, 31, 9	mndrid 18781	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) = 𝑏)
57	15, 26, 56	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑏(+g‘𝑆)(0g‘𝑆)) = 𝑏)
58	55, 57	opeq12d 4886	. . . . . . . . 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 2775	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7438	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩))
61	60, 42	eqeq12d 2751	. . . . . . . 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 3211	. . . . . 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 18694	. 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ = (0g‘𝑇))
68	67	eqcomd 2741	1 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  ⟨cop 4637   × cxp 5687  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486   ×_s cxps 17553  Mndcmnd 18760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-imas 17555  df-xps 17557  df-mgm 18666  df-sgrp 18745  df-mnd 18761

This theorem is referenced by:  xpsinv  19091  rngqiprngimf1  21328