Theorem lmod1 48481

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 2729	. . . . 5 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
2	1	grp1 18979	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3		fvex 6871	. . . . . . 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 5391	. . . . . . . . . . . . 13 ⊢ {𝐼} ∈ V
6	1	grpbase 17252	. . . . . . . . . . . . 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 4841	. . . . . . . . . . 11 ⊢ ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9		tpeq1 4706	. . . . . . . . . . 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 4127	. . . . . . . . 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 2752	. . . . . . . 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 17289	. . . . . . 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 2738	. . . . 5 ⊢ (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16		fvex 6871	. . . . . . 7 ⊢ (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17		snex 5391	. . . . . . . . . . . . 13 ⊢ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
18	1	grpplusg 17253	. . . . . . . . . . . . 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 4841	. . . . . . . . . . 11 ⊢ ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21		tpeq2 4707	. . . . . . . . . . 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 4127	. . . . . . . . 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 2752	. . . . . . . 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 17290	. . . . . . 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 2738	. . . . 5 ⊢ (+g‘𝑀) = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
28	15, 27	grpprop 18884	. . . 4 ⊢ (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
29	2, 28	sylibr 234	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp)
30	29	adantr 480	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ Grp)
31	4	lmodsca 17291	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
32	31	eqcomd 2735	. . . 4 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑀) = 𝑅)
33	32	adantl 481	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) = 𝑅)
34		simpr 484	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
35	33, 34	eqeltrd 2828	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
36	33	fveq2d 6862	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
37	36	eleq2d 2814	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
38	36	eleq2d 2814	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
39	37, 38	anbi12d 632	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) ↔ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))))
40		simpll 766	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼 ∈ 𝑉)
41		simplr 768	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
42		simprr 772	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅))
43	40, 41, 42	3jca 1128	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
44	4	lmod1lem1 48476	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼})
45	43, 44	syl 17	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼})
46	4	lmod1lem2 48477	. . . . . . . . 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 17	. . . . . . . 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 48478	. . . . . . . 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 1128	. . . . . . 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 48479	. . . . . . 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 48480	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)
52	51	adantr 480	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)
53	49, 50, 52	jca32 515	. . . . . 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 412	. . . . 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 240	. . . 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 3178	. . 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 7395	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑤(+g‘𝑀)𝑥) = (𝑤(+g‘𝑀)𝐼))
58	57	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)))
59		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)𝑥) = (𝑟( ·𝑠 ‘𝑀)𝐼))
60	59	oveq2d 7403	. . . . . . . . . . 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 2745	. . . . . . . . . 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 1443	. . . . . . . . 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 631	. . . . . . . 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 3156	. . . . . . 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 4639	. . . . . 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 480	. . . . 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 7395	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)𝑤) = (𝑟( ·𝑠 ‘𝑀)𝐼))
68	67	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼}))
69		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑤(+g‘𝑀)𝐼) = (𝐼(+g‘𝑀)𝐼))
70	69	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)))
71	67	oveq1d 7402	. . . . . . . . . 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 2745	. . . . . . . . 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 7395	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼))
74		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑞( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)𝐼))
75	74, 67	oveq12d 7405	. . . . . . . . . 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 2745	. . . . . . . . 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 1438	. . . . . . . 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 7395	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼))
79	67	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
80	78, 79	eqeq12d 2745	. . . . . . . . 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 7395	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼))
82		id 22	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → 𝑤 = 𝐼)
83	81, 82	eqeq12d 2745	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑤) = 𝑤 ↔ ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼))
84	80, 83	anbi12d 632	. . . . . . . 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 632	. . . . . . 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 4639	. . . . . 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 480	. . . . 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 279	. . . 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 3201	. . 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 257	. 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 17289	. . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
92	5, 91	ax-mp 5	. . 3 ⊢ {𝐼} = (Base‘𝑀)
93		eqid 2729	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
94		eqid 2729	. . 3 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
95		eqid 2729	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
96		eqid 2729	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97		eqid 2729	. . 3 ⊢ (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
98		eqid 2729	. . 3 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
99		eqid 2729	. . 3 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 20770	. 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 1344	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∪ cun 3912  {csn 4589  {cpr 4591  {ctp 4593  ⟨cop 4595  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ndxcnx 17163  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  Grpcgrp 18865  1_rcur 20090  Ringcrg 20142  LModclmod 20766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-sca 17236  df-vsca 17237  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-mgp 20050  df-ur 20091  df-ring 20144  df-lmod 20768

This theorem is referenced by:  lmod1zr  48482