lmod1 - Mathbox for Alexander van der Vekens

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Theorem lmod1 47551

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 2728	. . . . 5 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
2	1	grp1 18997	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3		fvex 6905	. . . . . . 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 5428	. . . . . . . . . . . . 13 ⊢ {𝐼} ∈ V
6	1	grpbase 17261	. . . . . . . . . . . . 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 4874	. . . . . . . . . . 11 ⊢ ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9		tpeq1 4743	. . . . . . . . . . 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 4156	. . . . . . . . 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 2756	. . . . . . . 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 17301	. . . . . . 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 2737	. . . . 5 ⊢ (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16		fvex 6905	. . . . . . 7 ⊢ (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17		snex 5428	. . . . . . . . . . . . 13 ⊢ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
18	1	grpplusg 17263	. . . . . . . . . . . . 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 4874	. . . . . . . . . . 11 ⊢ ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+_g‘ndx), (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21		tpeq2 4744	. . . . . . . . . . 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 4156	. . . . . . . . 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 2756	. . . . . . . 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 17302	. . . . . . 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 2737	. . . . 5 ⊢ (+_g‘𝑀) = (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
28	15, 27	grpprop 18903	. . . 4 ⊢ (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
29	2, 28	sylibr 233	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp)
30	29	adantr 480	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ Grp)
31	4	lmodsca 17303	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
32	31	eqcomd 2734	. . . 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 2829	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
36	33	fveq2d 6896	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
37	36	eleq2d 2815	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
38	36	eleq2d 2815	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
39	37, 38	anbi12d 631	. . . . 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 1126	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
44	4	lmod1lem1 47546	. . . . . . . . 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 47547	. . . . . . . . 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 47548	. . . . . . . 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 1126	. . . . . . 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 47549	. . . . . . 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 47550	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼)
52	51	adantr 480	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼))))
55	39, 54	sylbid 239	. . . 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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼))))
56	55	ralrimivv 3194	. . 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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼)))
57		oveq2 7423	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑤(+_g‘𝑀)𝑥) = (𝑤(+_g‘𝑀)𝐼))
58	57	oveq2d 7431	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝑥)) = (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝐼)))
59		oveq2 7423	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)𝑥) = (𝑟( ·_𝑠 ‘𝑀)𝐼))
60	59	oveq2d 7431	. . . . . . . . . . 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 2744	. . . . . . . . . 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 1438	. . . . . . . . 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 630	. . . . . . . 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤))))
64	63	ralbidv 3173	. . . . . . 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤))))
65	64	ralsng 4674	. . . . . 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤))))
67		oveq2 7423	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)𝑤) = (𝑟( ·_𝑠 ‘𝑀)𝐼))
68	67	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → ((𝑟( ·_𝑠 ‘𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·_𝑠 ‘𝑀)𝐼) ∈ {𝐼}))
69		oveq1 7422	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑤(+_g‘𝑀)𝐼) = (𝐼(+_g‘𝑀)𝐼))
70	69	oveq2d 7431	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝐼)) = (𝑟( ·_𝑠 ‘𝑀)(𝐼(+_g‘𝑀)𝐼)))
71	67	oveq1d 7430	. . . . . . . . . 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 2744	. . . . . . . . 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 7423	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(+_g‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = ((𝑞(+_g‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼))
74		oveq2 7423	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑞( ·_𝑠 ‘𝑀)𝑤) = (𝑞( ·_𝑠 ‘𝑀)𝐼))
75	74, 67	oveq12d 7433	. . . . . . . . . 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 2744	. . . . . . . . 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 1433	. . . . . . . 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 7423	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = ((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼))
79	67	oveq2d 7431	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑞( ·_𝑠 ‘𝑀)(𝑟( ·_𝑠 ‘𝑀)𝑤)) = (𝑞( ·_𝑠 ‘𝑀)(𝑟( ·_𝑠 ‘𝑀)𝐼)))
80	78, 79	eqeq12d 2744	. . . . . . . . 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 7423	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼))
82		id 22	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → 𝑤 = 𝐼)
83	81, 82	eqeq12d 2744	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤 ↔ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼))
84	80, 83	anbi12d 631	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → ((((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = (𝑞( ·_𝑠 ‘𝑀)(𝑟( ·_𝑠 ‘𝑀)𝑤)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤) ↔ (((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼) = (𝑞( ·_𝑠 ‘𝑀)(𝑟( ·_𝑠 ‘𝑀)𝐼)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼)))
85	77, 84	anbi12d 631	. . . . . . 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼))))
86	85	ralsng 4674	. . . . . 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼))))
89	88	2ralbidv 3214	. . 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤)))
91	4	lmodbase 17301	. . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
92	5, 91	ax-mp 5	. . 3 ⊢ {𝐼} = (Base‘𝑀)
93		eqid 2728	. . 3 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
94		eqid 2728	. . 3 ⊢ ( ·_𝑠 ‘𝑀) = ( ·_𝑠 ‘𝑀)
95		eqid 2728	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
96		eqid 2728	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97		eqid 2728	. . 3 ⊢ (+_g‘(Scalar‘𝑀)) = (+_g‘(Scalar‘𝑀))
98		eqid 2728	. . 3 ⊢ (._r‘(Scalar‘𝑀)) = (._r‘(Scalar‘𝑀))
99		eqid 2728	. . 3 ⊢ (1_r‘(Scalar‘𝑀)) = (1_r‘(Scalar‘𝑀))
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 20741	. 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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤))))
101	30, 35, 90, 100	syl3anbrc 1341	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∪ cun 3943 {csn 4625 {cpr 4627 {ctp 4629 ⟨cop 4631 ‘cfv 6543 (class class class)co 7415 ∈ cmpo 7417 ndxcnx 17156 Basecbs 17174 +_gcplusg 17227 ._rcmulr 17228 Scalarcsca 17230 ·_𝑠 cvsca 17231 Grpcgrp 18884 1_rcur 20115 Ringcrg 20167 LModclmod 20737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-sca 17243 df-vsca 17244 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-mgp 20069 df-ur 20116 df-ring 20169 df-lmod 20739

This theorem is referenced by: lmod1zr 47552

W3C validator