Theorem xpsmnd0 18742

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 2728	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
2		eqid 2728	. . 3 ⊢ (0g‘𝑇) = (0g‘𝑇)
3		eqid 2728	. . 3 ⊢ (+g‘𝑇) = (+g‘𝑇)
4		eqid 2728	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2728	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
6	4, 5	mndidcl 18716	. . . . . 6 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅))
7	6	adantr 479	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑅) ∈ (Base‘𝑅))
8		eqid 2728	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2728	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	8, 9	mndidcl 18716	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ (Base‘𝑆))
11	10	adantl 480	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑆) ∈ (Base‘𝑆))
12	7, 11	opelxpd 5721	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ ∈ ((Base‘𝑅) × (Base‘𝑆)))
13		xpsmnd0.t	. . . . 5 ⊢ 𝑇 = (𝑅 ×s 𝑆)
14		simpl 481	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑅 ∈ Mnd)
15		simpr 483	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑆 ∈ Mnd)
16	13, 4, 8, 14, 15	xpsbas 17561	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ((Base‘𝑅) × (Base‘𝑆)) = (Base‘𝑇))
17	12, 16	eleqtrd 2831	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ ∈ (Base‘𝑇))
18	16	eleq2d 2815	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ 𝑥 ∈ (Base‘𝑇)))
19		elxp2 5706	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) ↔ ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ (Base‘𝑆)𝑥 = ⟨𝑎, 𝑏⟩)
20	14	adantr 479	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑅 ∈ Mnd)
21	15	adantr 479	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑆 ∈ Mnd)
22	7	adantr 479	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (0g‘𝑅) ∈ (Base‘𝑅))
23	11	adantr 479	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (0g‘𝑆) ∈ (Base‘𝑆))
24		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆)) → 𝑎 ∈ (Base‘𝑅))
25	24	adantl 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑎 ∈ (Base‘𝑅))
26		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆)) → 𝑏 ∈ (Base‘𝑆))
27	26	adantl 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → 𝑏 ∈ (Base‘𝑆))
28		eqid 2728	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
29	4, 28	mndcl 18709	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
30	20, 22, 25, 29	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
31		eqid 2728	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	8, 31	mndcl 18709	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ (0g‘𝑆) ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	21, 23, 27, 32	syl3anc 1368	. . . . . . . . . 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 17563	. . . . . . . . 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 18721	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) = 𝑎)
36	14, 24, 35	syl2an 594	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) = 𝑎)
37	8, 31, 9	mndlid 18721	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) = 𝑏)
38	15, 26, 37	syl2an 594	. . . . . . . . . 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 2768	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
41		oveq2 7434	. . . . . . . . 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 2744	. . . . . . . 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 246	. . . . . . 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 3209	. . . . . 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 241	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥))
47	18, 46	sylbird 259	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ (Base‘𝑇) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥))
48	47	imp 405	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝑇)) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)𝑥) = 𝑥)
49	4, 28	mndcl 18709	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅) ∧ (0g‘𝑅) ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
50	20, 25, 22, 49	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
51	8, 31	mndcl 18709	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) ∈ (Base‘𝑆))
52	21, 27, 23, 51	syl3anc 1368	. . . . . . . . . 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 17563	. . . . . . . . 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 18722	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) = 𝑎)
55	14, 24, 54	syl2an 594	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) = 𝑎)
56	8, 31, 9	mndrid 18722	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) = 𝑏)
57	15, 26, 56	syl2an 594	. . . . . . . . . 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 2768	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7433	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩))
61	60, 42	eqeq12d 2744	. . . . . . . 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 246	. . . . . . 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 3209	. . . . . 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 241	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ ((Base‘𝑅) × (Base‘𝑆)) → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥))
65	18, 64	sylbird 259	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑥 ∈ (Base‘𝑇) → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥))
66	65	imp 405	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ 𝑥 ∈ (Base‘𝑇)) → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = 𝑥)
67	1, 2, 3, 17, 48, 66	ismgmid2 18635	. 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ = (0g‘𝑇))
68	67	eqcomd 2734	1 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3067  ⟨cop 4638   × cxp 5680  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  +_gcplusg 17240  0_gc0g 17428   ×_s cxps 17495  Mndcmnd 18701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-hom 17264  df-cco 17265  df-0g 17430  df-prds 17436  df-imas 17497  df-xps 17499  df-mgm 18607  df-sgrp 18686  df-mnd 18702

This theorem is referenced by:  xpsinv  19023  rngqiprngimf1  21197