lmod1 - Mathbox for Alexander van der Vekens

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Theorem lmod1 48478

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 18944	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3		fvex 6839	. . . . . . 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 5378	. . . . . . . . . . . . 13 ⊢ {𝐼} ∈ V
6	1	grpbase 17211	. . . . . . . . . . . . 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 4831	. . . . . . . . . . 11 ⊢ ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9		tpeq1 4696	. . . . . . . . . . 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 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 17248	. . . . . . 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 6839	. . . . . . 7 ⊢ (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17		snex 5378	. . . . . . . . . . . . 13 ⊢ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
18	1	grpplusg 17212	. . . . . . . . . . . . 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 4831	. . . . . . . . . . 11 ⊢ ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+_g‘ndx), (+_g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+_g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21		tpeq2 4697	. . . . . . . . . . 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 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 17249	. . . . . . 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 18849	. . . 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 17250	. . . . 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 6830	. . . . . . 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 48473	. . . . . . . . 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 48474	. . . . . . . . 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 48475	. . . . . . . 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 48476	. . . . . . 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 48477	. . . . . . . 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 3170	. . 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 7361	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑤(+_g‘𝑀)𝑥) = (𝑤(+_g‘𝑀)𝐼))
58	57	oveq2d 7369	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝑥)) = (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝐼)))
59		oveq2 7361	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)𝑥) = (𝑟( ·_𝑠 ‘𝑀)𝐼))
60	59	oveq2d 7369	. . . . . . . . . . 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)) ∧ ((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 3152	. . . . . . 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 4629	. . . . . 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 7361	. . . . . . . . . 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 7360	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑤(+_g‘𝑀)𝐼) = (𝐼(+_g‘𝑀)𝐼))
70	69	oveq2d 7369	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → (𝑟( ·_𝑠 ‘𝑀)(𝑤(+_g‘𝑀)𝐼)) = (𝑟( ·_𝑠 ‘𝑀)(𝐼(+_g‘𝑀)𝐼)))
71	67	oveq1d 7368	. . . . . . . . . 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 7361	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(+_g‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = ((𝑞(+_g‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼))
74		oveq2 7361	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐼 → (𝑞( ·_𝑠 ‘𝑀)𝑤) = (𝑞( ·_𝑠 ‘𝑀)𝐼))
75	74, 67	oveq12d 7371	. . . . . . . . . 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 7361	. . . . . . . . . 10 ⊢ (𝑤 = 𝐼 → ((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝑤) = ((𝑞(._r‘(Scalar‘𝑀))𝑟)( ·_𝑠 ‘𝑀)𝐼))
79	67	oveq2d 7369	. . . . . . . . . 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 7361	. . . . . . . . . 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 2745	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (((1_r‘(Scalar‘𝑀))( ·_𝑠 ‘𝑀)𝑤) = 𝑤 ↔ ((1_r‘(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)) ∧ ((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 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)) ∧ ((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 4629	. . . . . 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 3193	. . 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 17248	. . . 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 ⊢ (1_r‘(Scalar‘𝑀)) = (1_r‘(Scalar‘𝑀))
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 20785	. 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 1344	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3438 ∪ cun 3903 {csn 4579 {cpr 4581 {ctp 4583 ⟨cop 4585 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 ndxcnx 17122 Basecbs 17138 +_gcplusg 17179 ._rcmulr 17180 Scalarcsca 17182 ·_𝑠 cvsca 17183 Grpcgrp 18830 1_rcur 20084 Ringcrg 20136 LModclmod 20781

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-n0 12403 df-z 12490 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-sca 17195 df-vsca 17196 df-0g 17363 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-grp 18833 df-mgp 20044 df-ur 20085 df-ring 20138 df-lmod 20783

This theorem is referenced by: lmod1zr 48479

W3C validator