lmod1 - Mathbox for Alexander van der Vekens

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Theorem lmod1 48968

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 2736	. . . . 5 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
2	1	grp1 19023	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3		fvex 6853	. . . . . . 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 5381	. . . . . . . . . . . . 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 4820	. . . . . . . . . . 11 ⊢ ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9		tpeq1 4686	. . . . . . . . . . 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 4104	. . . . . . . . 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 2759	. . . . . . . 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 2745	. . . . 5 ⊢ (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16		fvex 6853	. . . . . . 7 ⊢ (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17		snex 5381	. . . . . . . . . . . . 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 4820	. . . . . . . . . . 11 ⊢ ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+_g‘ndx), (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21		tpeq2 4687	. . . . . . . . . . 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 4104	. . . . . . . . 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 2759	. . . . . . . 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 2745	. . . . 5 ⊢ (+_g‘𝑀) = (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
28	15, 27	grpprop 18928	. . . 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 2742	. . . 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 2836	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
36	33	fveq2d 6844	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
37	36	eleq2d 2822	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
38	36	eleq2d 2822	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
39	37, 38	anbi12d 633	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) ↔ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))))
40		simpll 767	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼 ∈ 𝑉)
41		simplr 769	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
42		simprr 773	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅))
43	40, 41, 42	3jca 1129	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
44	4	lmod1lem1 48963	. . . . . . . . 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 48964	. . . . . . . . 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 48965	. . . . . . . 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 1129	. . . . . . 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 48966	. . . . . . 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 48967	. . . . . . . 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 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)) ∧ ((1_r‘(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)) ∧ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼)))
57		oveq2 7375	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑤(+_g‘𝑀)𝑥) = (𝑤(+_g‘𝑀)𝐼))
58	57	oveq2d 7383	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝑥)) = (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝐼)))
59		oveq2 7375	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)𝑥) = (𝑟( ·_𝑠 ‘𝑀)𝐼))
60	59	oveq2d 7383	. . . . . . . . . . 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 2752	. . . . . . . . . 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 1444	. . . . . . . . 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 632	. . . . . . . 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 3160	. . . . . . 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 4619	. . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)𝑤) = (𝑟( ·_𝑠 ‘𝑀)𝐼))
68	67	eleq1d 2821	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → ((𝑟( ·_𝑠 ‘𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·_𝑠 ‘𝑀)𝐼) ∈ {𝐼}))
69		oveq1 7374	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑤(+_g‘𝑀)𝐼) = (𝐼(+_g‘𝑀)𝐼))
70	69	oveq2d 7383	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝐼)) = (𝑟( ·_𝑠 ‘𝑀)(𝐼(+_g‘𝑀)𝐼)))
71	67	oveq1d 7382	. . . . . . . . . 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 2752	. . . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(+_g‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = ((𝑞(+_g‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼))
74		oveq2 7375	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑞( ·_𝑠 ‘𝑀)𝑤) = (𝑞( ·_𝑠 ‘𝑀)𝐼))
75	74, 67	oveq12d 7385	. . . . . . . . . 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 2752	. . . . . . . . 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 1439	. . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = ((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼))
79	67	oveq2d 7383	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑞( ·_𝑠 ‘𝑀)(𝑟( ·_𝑠 ‘𝑀)𝑤)) = (𝑞( ·_𝑠 ‘𝑀)(𝑟( ·_𝑠 ‘𝑀)𝐼)))
80	78, 79	eqeq12d 2752	. . . . . . . . 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 7375	. . . . . . . . . 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 2752	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤 ↔ ((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝐼) = 𝐼))
84	80, 83	anbi12d 633	. . . . . . . 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 633	. . . . . . 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 4619	. . . . . 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 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)) ∧ ((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 17289	. . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
92	5, 91	ax-mp 5	. . 3 ⊢ {𝐼} = (Base‘𝑀)
93		eqid 2736	. . 3 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
94		eqid 2736	. . 3 ⊢ ( ·_𝑠 ‘𝑀) = ( ·_𝑠 ‘𝑀)
95		eqid 2736	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
96		eqid 2736	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97		eqid 2736	. . 3 ⊢ (+_g‘(Scalar‘𝑀)) = (+_g‘(Scalar‘𝑀))
98		eqid 2736	. . 3 ⊢ (._r‘(Scalar‘𝑀)) = (._r‘(Scalar‘𝑀))
99		eqid 2736	. . 3 ⊢ (1_r‘(Scalar‘𝑀)) = (1_r‘(Scalar‘𝑀))
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 20859	. 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 1345	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 Vcvv 3429 ∪ cun 3887 {csn 4567 {cpr 4569 {ctp 4571 ⟨cop 4573 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ndxcnx 17163 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 Scalarcsca 17223 ·_𝑠 cvsca 17224 Grpcgrp 18909 1_rcur 20162 Ringcrg 20214 LModclmod 20855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 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 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-mgp 20122 df-ur 20163 df-ring 20216 df-lmod 20857

This theorem is referenced by: lmod1zr 48969

W3C validator