Theorem xpsmnd0 18703

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 2736	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
2		eqid 2736	. . 3 ⊢ (0g‘𝑇) = (0g‘𝑇)
3		eqid 2736	. . 3 ⊢ (+g‘𝑇) = (+g‘𝑇)
4		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
6	4, 5	mndidcl 18674	. . . . . 6 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅))
7	6	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑅) ∈ (Base‘𝑅))
8		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	8, 9	mndidcl 18674	. . . . . 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 17493	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ((Base‘𝑅) × (Base‘𝑆)) = (Base‘𝑇))
17	12, 16	eleqtrd 2838	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ ∈ (Base‘𝑇))
18	16	eleq2d 2822	. . . . 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 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝑅) = (+g‘𝑅)
29	4, 28	mndcl 18667	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
30	20, 22, 25, 29	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → ((0g‘𝑅)(+g‘𝑅)𝑎) ∈ (Base‘𝑅))
31		eqid 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	8, 31	mndcl 18667	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ (0g‘𝑆) ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → ((0g‘𝑆)(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	21, 23, 27, 32	syl3anc 1373	. . . . . . . . . 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 17495	. . . . . . . . 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 18679	. . . . . . . . . . 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 18679	. . . . . . . . . . 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 4837	. . . . . . . . 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 2771	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨(0g‘𝑅), (0g‘𝑆)⟩(+g‘𝑇)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
41		oveq2 7366	. . . . . . . . 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 2752	. . . . . . . 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 3193	. . . . . 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 18667	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ 𝑎 ∈ (Base‘𝑅) ∧ (0g‘𝑅) ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
50	20, 25, 22, 49	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (𝑎(+g‘𝑅)(0g‘𝑅)) ∈ (Base‘𝑅))
51	8, 31	mndcl 18667	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑏 ∈ (Base‘𝑆) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → (𝑏(+g‘𝑆)(0g‘𝑆)) ∈ (Base‘𝑆))
52	21, 27, 23, 51	syl3anc 1373	. . . . . . . . . 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 17495	. . . . . . . . 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 18680	. . . . . . . . . . 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 18680	. . . . . . . . . . 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 4837	. . . . . . . . 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 2771	. . . . . . . 8 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑆))) → (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7365	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩) = (⟨𝑎, 𝑏⟩(+g‘𝑇)⟨(0g‘𝑅), (0g‘𝑆)⟩))
61	60, 42	eqeq12d 2752	. . . . . . . 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 3193	. . . . . 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 18593	. 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → ⟨(0g‘𝑅), (0g‘𝑆)⟩ = (0g‘𝑇))
68	67	eqcomd 2742	1 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ⟨cop 4586   × cxp 5622  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  0_gc0g 17359   ×_s cxps 17427  Mndcmnd 18659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-prds 17367  df-imas 17429  df-xps 17431  df-mgm 18565  df-sgrp 18644  df-mnd 18660

This theorem is referenced by:  xpsinv  18990  rngqiprngimf1  21255