Theorem lmod1 49300

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 2762	. . . . 5 ⊢ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
2	1	grp1 19119	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3		fvex 6894	. . . . . . 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 5409	. . . . . . . . . . . . 13 ⊢ {𝐼} ∈ V
6	1	grpbase 17348	. . . . . . . . . . . . 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 4707	. . . . . . . . . . 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 4117	. . . . . . . . 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 2785	. . . . . . . 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 17385	. . . . . . 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 2771	. . . . 5 ⊢ (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16		fvex 6894	. . . . . . 7 ⊢ (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17		snex 5409	. . . . . . . . . . . . 13 ⊢ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
18	1	grpplusg 17349	. . . . . . . . . . . . 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 4708	. . . . . . . . . . 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 4117	. . . . . . . . 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 2785	. . . . . . . 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 17386	. . . . . . 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 2771	. . . . 5 ⊢ (+g‘𝑀) = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
28	15, 27	grpprop 19025	. . . 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 17387	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
32	31	eqcomd 2768	. . . 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 2862	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
36	33	fveq2d 6885	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
37	36	eleq2d 2848	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
38	36	eleq2d 2848	. . . . . 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 1145	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
44	4	lmod1lem1 49295	. . . . . . . . 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 49296	. . . . . . . . 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 49297	. . . . . . . 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 1145	. . . . . . 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 49298	. . . . . . 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 49299	. . . . . . . 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 3205	. . 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 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑤(+g‘𝑀)𝑥) = (𝑤(+g‘𝑀)𝐼))
58	57	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝑥)) = (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)))
59		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)𝑥) = (𝑟( ·𝑠 ‘𝑀)𝐼))
60	59	oveq2d 7428	. . . . . . . . . . 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 2778	. . . . . . . . . 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 1468	. . . . . . . . 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 3187	. . . . . . 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 4640	. . . . . 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 7420	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)𝑤) = (𝑟( ·𝑠 ‘𝑀)𝐼))
68	67	eleq1d 2847	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → ((𝑟( ·𝑠 ‘𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼}))
69		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑤(+g‘𝑀)𝐼) = (𝐼(+g‘𝑀)𝐼))
70	69	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·𝑠 ‘𝑀)(𝑤(+g‘𝑀)𝐼)) = (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)))
71	67	oveq1d 7427	. . . . . . . . . 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 2778	. . . . . . . . 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 7420	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼))
74		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑞( ·𝑠 ‘𝑀)𝑤) = (𝑞( ·𝑠 ‘𝑀)𝐼))
75	74, 67	oveq12d 7430	. . . . . . . . . 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 2778	. . . . . . . . 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 1463	. . . . . . . 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 7420	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝑤) = ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼))
79	67	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝑤)) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
80	78, 79	eqeq12d 2778	. . . . . . . . 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 7420	. . . . . . . . . 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 2778	. . . . . . . . 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 4640	. . . . . 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 3228	. . 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 17385	. . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
92	5, 91	ax-mp 5	. . 3 ⊢ {𝐼} = (Base‘𝑀)
93		eqid 2762	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
94		eqid 2762	. . 3 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
95		eqid 2762	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
96		eqid 2762	. . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97		eqid 2762	. . 3 ⊢ (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
98		eqid 2762	. . 3 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
99		eqid 2762	. . 3 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 20996	. 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 1361	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ∪ cun 3902  {csn 4588  {cpr 4590  {ctp 4592  ⟨cop 4594  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  ndxcnx 17259  Basecbs 17275  +_gcplusg 17316  ._rcmulr 17317  Scalarcsca 17319   ·_𝑠 cvsca 17320  Grpcgrp 19006  1_rcur 20269  Ringcrg 20321  LModclmod 20992

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-n0 12511  df-z 12598  df-uz 12869  df-fz 13542  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-plusg 17329  df-sca 17332  df-vsca 17333  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-grp 19009  df-mgp 20223  df-ur 20270  df-ring 20323  df-lmod 20994

This theorem is used by:  lmod1zr  49301