Theorem lmod1 49123

Description: The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)

Hypothesis

Ref	Expression
lmod1.m	⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})

Assertion

Ref	Expression
lmod1	⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Distinct variable groups:   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥,𝑉,𝑦   𝑥,𝑀,𝑦

Proof of Theorem lmod1

Dummy variables 𝑟 𝑞 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
2	1	grp1 19104	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3		fvex 6884	. . . . . . 7 ⊢ (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
4		lmod1.m	. . . . . . . . 9 ⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
5		snex 5401	. . . . . . . . . . . . 13 ⊢ {𝐼} ∈ V
6	1	grpbase 17332	. . . . . . . . . . . . 13 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}))
7	5, 6	ax-mp 5	. . . . . . . . . . . 12 ⊢ {𝐼} = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
8	7	opeq2i 4838	. . . . . . . . . . 11 ⊢ ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9		tpeq1 4704	. . . . . . . . . . 11 ⊢ (⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩ → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩})
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩}
11	10	uneq1i 4120	. . . . . . . . 9 ⊢ ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩}) = ({⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
12	4, 11	eqtri 2788	. . . . . . . 8 ⊢ 𝑀 = ({⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
13	12	lmodbase 17369	. . . . . . 7 ⊢ ((Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V → (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (Base‘𝑀))
14	3, 13	ax-mp 5	. . . . . 6 ⊢ (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (Base‘𝑀)
15	14	eqcomi 2774	. . . . 5 ⊢ (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16		fvex 6884	. . . . . . 7 ⊢ (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17		snex 5401	. . . . . . . . . . . . 13 ⊢ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
18	1	grpplusg 17333	. . . . . . . . . . . . 13 ⊢ ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}))
19	17, 18	ax-mp 5	. . . . . . . . . . . 12 ⊢ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
20	19	opeq2i 4838	. . . . . . . . . . 11 ⊢ ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21		tpeq2 4705	. . . . . . . . . . 11 ⊢ (⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩ → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩})
22	20, 21	ax-mp 5	. . . . . . . . . 10 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩}
23	22	uneq1i 4120	. . . . . . . . 9 ⊢ ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩}) = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
24	4, 23	eqtri 2788	. . . . . . . 8 ⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
25	24	lmodplusg 17370	. . . . . . 7 ⊢ ((+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V → (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (+g‘𝑀))
26	16, 25	ax-mp 5	. . . . . 6 ⊢ (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (+g‘𝑀)
27	26	eqcomi 2774	. . . . 5 ⊢ (+g‘𝑀) = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
28	15, 27	grpprop 19009	. . . 4 ⊢ (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
29	2, 28	sylibr 237	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp)
30	29	adantr 485	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ Grp)
31	4	lmodsca 17371	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
32	31	eqcomd 2771	. . . 4 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑀) = 𝑅)
33	32	adantl 486	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) = 𝑅)
34		simpr 489	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
35	33, 34	eqeltrd 2865	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
36	33	fveq2d 6875	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
37	36	eleq2d 2851	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
38	36	eleq2d 2851	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
39	37, 38	anbi12d 643	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) ↔ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))))
40		simpll 778	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼 ∈ 𝑉)
41		simplr 780	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
42		simprr 784	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅))
43	40, 41, 42	3jca 1144	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
44	4	lmod1lem1 49118	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼})
45	43, 44	syl 18	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼})
46	4	lmod1lem2 49119	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
47	43, 46	syl 18	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
48	4	lmod1lem3 49120	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
49	45, 47, 48	3jca 1144	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))))
50	4	lmod1lem4 49121	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
51	4	lmod1lem5 49122	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)
52	51	adantr 485	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)
53	49, 50, 52	jca32 524	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)))
54	53	ex 417	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅)) → (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
55	39, 54	sylbid 243	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) → (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
56	55	ralrimivv 3206	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))(((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)))
57		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑤(+g‘𝑀)𝑥) = (𝑤(+g‘𝑀)𝐼))
58	57	oveq2d 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)))
59		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)𝑥) = (𝑟( ·𝑠 ‘𝑀)𝐼))
60	59	oveq2d 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
61	58, 60	eqeq12d 2781	. . . . . . . . . 10 ⊢ (𝑥 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ↔ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))))
62	61	3anbi2d 1465	. . . . . . . . 9 ⊢ (𝑥 = 𝐼 → (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ↔ ((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)))))
63	62	anbi1d 642	. . . . . . . 8 ⊢ (𝑥 = 𝐼 → ((((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤))))
64	63	ralbidv 3188	. . . . . . 7 ⊢ (𝑥 = 𝐼 → (∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤))))
65	64	ralsng 4637	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤))))
66	65	adantr 485	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤))))
67		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)𝑤) = (𝑟( ·𝑠 ‘𝑀)𝐼))
68	67	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼}))
69		oveq1 7407	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑤(+g‘𝑀)𝐼) = (𝐼(+g‘𝑀)𝐼))
70	69	oveq2d 7416	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)))
71	67	oveq1d 7415	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
72	70, 71	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ↔ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))))
73		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼))
74		oveq2 7408	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑞( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)𝐼))
75	74, 67	oveq12d 7418	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
76	73, 75	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ↔ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))))
77	68, 72, 76	3anbi123d 1460	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ↔ ((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))))
78		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼))
79	67	oveq2d 7416	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
80	78, 79	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ↔ ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))))
81		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼))
82		id 23	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → 𝑤 = 𝐼)
83	81, 82	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤 ↔ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))
84	80, 83	anbi12d 643	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → ((((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤) ↔ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)))
85	77, 84	anbi12d 643	. . . . . . 7 ⊢ (𝑤 = 𝐼 → ((((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
86	85	ralsng 4637	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
87	86	adantr 485	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
88	66, 87	bitrd 282	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
89	88	2ralbidv 3229	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))(((𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))))
90	56, 89	mpbird 260	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤)))
91	4	lmodbase 17369	. . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
92	5, 91	ax-mp 5	. . 3 ⊢ {𝐼} = (Base‘𝑀)
93		eqid 2765	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
94		eqid 2765	. . 3 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
95		eqid 2765	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
96		eqid 2765	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97		eqid 2765	. . 3 ⊢ (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
98		eqid 2765	. . 3 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
99		eqid 2765	. . 3 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 20954	. 2 ⊢ (𝑀 ∈ LMod ↔ (𝑀 ∈ Grp ∧ (Scalar‘𝑀) ∈ Ring ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = ((𝑟( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞( ·𝑠 ‘𝑀)𝑤)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤))))
101	30, 35, 90, 100	syl3anbrc 1360	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ∪ cun 3905  {csn 4585  {cpr 4587  {ctp 4589  ⟨cop 4591  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ndxcnx 17243  Basecbs 17259  +_gcplusg 17300  ._rcmulr 17301  Scalarcsca 17303   ·_𝑠 cvsca 17304  Grpcgrp 18990  1_rcur 20254  Ringcrg 20306  LModclmod 20950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-plusg 17313  df-sca 17316  df-vsca 17317  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-mgp 20208  df-ur 20255  df-ring 20308  df-lmod 20952

This theorem is referenced by:  lmod1zr  49124