Theorem frlmsnic 39453

Description: Given a free module with a singleton as the index set, that is, a free module of one-dimensional vectors, the function that maps each vector to its coordinate is a module isomorphism from that module to its ring of scalars seen as a module. (Contributed by Steven Nguyen, 18-Aug-2023.)

Hypotheses

Ref	Expression
frlmsnic.w	⊢ 𝑊 = (𝐾 freeLMod {𝐼})
frlmsnic.1	⊢ 𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼))

Assertion

Ref	Expression
frlmsnic	⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

Distinct variable groups:   𝑥,𝐼   𝑥,𝐾   𝑥,𝐹   𝑥,𝑊

Proof of Theorem frlmsnic

Dummy variables 𝑦 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2798	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2798	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
3		eqid 2798	. . 3 ⊢ ( ·𝑠 ‘(ringLMod‘𝐾)) = ( ·𝑠 ‘(ringLMod‘𝐾))
4		eqid 2798	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2798	. . 3 ⊢ (Scalar‘(ringLMod‘𝐾)) = (Scalar‘(ringLMod‘𝐾))
6		eqid 2798	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7		snex 5297	. . . . 5 ⊢ {𝐼} ∈ V
8		frlmsnic.w	. . . . . 6 ⊢ 𝑊 = (𝐾 freeLMod {𝐼})
9	8	frlmlmod 20438	. . . . 5 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝑊 ∈ LMod)
10	7, 9	mpan2 690	. . . 4 ⊢ (𝐾 ∈ Ring → 𝑊 ∈ LMod)
11	10	adantr 484	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
12		rlmlmod 19970	. . . 4 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ LMod)
13	12	adantr 484	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ LMod)
14		rlmsca 19965	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
15	14	adantr 484	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
16	8	frlmsca 20442	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝐾 = (Scalar‘𝑊))
17	7, 16	mpan2 690	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑊))
18	17	adantr 484	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘𝑊))
19	15, 18	eqtr3d 2835	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘(ringLMod‘𝐾)) = (Scalar‘𝑊))
20		rlmbas 19960	. . . 4 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾))
21		eqid 2798	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
22		rlmplusg 19961	. . . 4 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾))
23		lmodgrp 19634	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
24	11, 23	syl 17	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ Grp)
25		lmodgrp 19634	. . . . . 6 ⊢ ((ringLMod‘𝐾) ∈ LMod → (ringLMod‘𝐾) ∈ Grp)
26	12, 25	syl 17	. . . . 5 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ Grp)
27	26	adantr 484	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ Grp)
28		eqid 2798	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
29	8, 28, 1	frlmbasf 20449	. . . . . . . 8 ⊢ (({𝐼} ∈ V ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
30	7, 29	mpan 689	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑊) → 𝑥:{𝐼}⟶(Base‘𝐾))
31	30	adantl 485	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
32		snidg 4559	. . . . . . . 8 ⊢ (𝐼 ∈ V → 𝐼 ∈ {𝐼})
33	32	adantl 485	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐼 ∈ {𝐼})
34	33	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝐼 ∈ {𝐼})
35	31, 34	ffvelrnd 6829	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥‘𝐼) ∈ (Base‘𝐾))
36		frlmsnic.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼))
37	35, 36	fmptd 6855	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)⟶(Base‘𝐾))
38		simpll 766	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐾 ∈ Ring)
39	7	a1i 11	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → {𝐼} ∈ V)
40		simprl 770	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41		simprr 772	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
42	33	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
43		eqid 2798	. . . . . 6 ⊢ (+g‘𝐾) = (+g‘𝐾)
44	8, 1, 38, 39, 40, 41, 42, 43, 21	frlmvplusgvalc 20456	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥(+g‘𝑊)𝑦)‘𝐼) = ((𝑥‘𝐼)(+g‘𝐾)(𝑦‘𝐼)))
45	11	adantr 484	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
46	1, 21	lmodvacl 19641	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
47	45, 40, 41, 46	syl3anc 1368	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
48		fveq1 6644	. . . . . . 7 ⊢ (𝑡 = (𝑥(+g‘𝑊)𝑦) → (𝑡‘𝐼) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
49		fveq1 6644	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥‘𝐼) = (𝑡‘𝐼))
50	49	cbvmptv 5133	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
51	36, 50	eqtri 2821	. . . . . . 7 ⊢ 𝐹 = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
52		fvexd 6660	. . . . . . 7 ⊢ (𝑡 ∈ (Base‘𝑊) → (𝑡‘𝐼) ∈ V)
53	48, 51, 52	fvmpt3 6749	. . . . . 6 ⊢ ((𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊) → (𝐹‘(𝑥(+g‘𝑊)𝑦)) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
54	47, 53	syl 17	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥(+g‘𝑊)𝑦)) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
55	36	a1i 11	. . . . . . . 8 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)))
56		fvexd 6660	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥‘𝐼) ∈ V)
57	55, 56	fvmpt2d 6758	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝐹‘𝑥) = (𝑥‘𝐼))
58	40, 57	mpdan 686	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑥) = (𝑥‘𝐼))
59		fveq1 6644	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝐼) = (𝑦‘𝐼))
60		fvexd 6660	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) → (𝑥‘𝐼) ∈ V)
61	59, 36, 60	fvmpt3 6749	. . . . . . 7 ⊢ (𝑦 ∈ (Base‘𝑊) → (𝐹‘𝑦) = (𝑦‘𝐼))
62	41, 61	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑦) = (𝑦‘𝐼))
63	58, 62	oveq12d 7153	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝐹‘𝑥)(+g‘𝐾)(𝐹‘𝑦)) = ((𝑥‘𝐼)(+g‘𝐾)(𝑦‘𝐼)))
64	44, 54, 63	3eqtr4d 2843	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥(+g‘𝑊)𝑦)) = ((𝐹‘𝑥)(+g‘𝐾)(𝐹‘𝑦)))
65	1, 20, 21, 22, 24, 27, 37, 64	isghmd 18359	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 GrpHom (ringLMod‘𝐾)))
66	7	a1i 11	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → {𝐼} ∈ V)
67	18	eqcomd 2804	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘𝑊) = 𝐾)
68	67	fveq2d 6649	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Base‘(Scalar‘𝑊)) = (Base‘𝐾))
69	68	eleq2d 2875	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↔ 𝑥 ∈ (Base‘𝐾)))
70	69	biimpa 480	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
71	70	adantrr 716	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
72		simprr 772	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
73	33	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
74		eqid 2798	. . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾)
75	8, 1, 28, 66, 71, 72, 73, 2, 74	frlmvscaval 20457	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) = (𝑥(.r‘𝐾)(𝑦‘𝐼)))
76		rlmvsca 19967	. . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾))
77	76	oveqi 7148	. . . . 5 ⊢ (𝑥(.r‘𝐾)(𝑦‘𝐼)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼))
78	75, 77	eqtrdi 2849	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼)))
79		fveq1 6644	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥‘𝐼) = (𝑢‘𝐼))
80	79	cbvmptv 5133	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
81	36, 80	eqtri 2821	. . . . 5 ⊢ 𝐹 = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
82		fveq1 6644	. . . . 5 ⊢ (𝑢 = (𝑥( ·𝑠 ‘𝑊)𝑦) → (𝑢‘𝐼) = ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼))
83	7	a1i 11	. . . . . . . 8 ⊢ (𝐼 ∈ V → {𝐼} ∈ V)
84	83, 9	sylan2 595	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
85	84	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
86		simprl 770	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
87	1, 4, 2, 6	lmodvscl 19644	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
88	85, 86, 72, 87	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
89		fvexd 6660	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) ∈ V)
90	81, 82, 88, 89	fvmptd3 6768	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼))
91		fvex 6658	. . . . . . 7 ⊢ (𝑥‘𝐼) ∈ V
92	59, 36, 91	fvmpt3i 6750	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑊) → (𝐹‘𝑦) = (𝑦‘𝐼))
93	72, 92	syl 17	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑦) = (𝑦‘𝐼))
94	93	oveq2d 7151	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼)))
95	78, 90, 94	3eqtr4d 2843	. . 3 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝐹‘𝑦)))
96	1, 2, 3, 4, 5, 6, 11, 13, 19, 65, 95	islmhmd 19804	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMHom (ringLMod‘𝐾)))
97		simplr 768	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐼 ∈ V)
98		simpr 488	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
99	97, 98	fsnd 6632	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾))
100		simpll 766	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐾 ∈ Ring)
101		snfi 8577	. . . . . 6 ⊢ {𝐼} ∈ Fin
102	8, 28, 1	frlmfielbas 39434	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ Fin) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
103	100, 101, 102	sylancl 589	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
104	99, 103	mpbird 260	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊))
105		fveq1 6644	. . . . . . . 8 ⊢ (𝑥 = {⟨𝐼, 𝑦⟩} → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
106	105	adantl 485	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
107		simpllr 775	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝐼 ∈ V)
108		vex 3444	. . . . . . . 8 ⊢ 𝑦 ∈ V
109		fvsng 6919	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑦 ∈ V) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
110	107, 108, 109	sylancl 589	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
111	106, 110	eqtr2d 2834	. . . . . 6 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝑦 = (𝑥‘𝐼))
112	111	ex 416	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} → 𝑦 = (𝑥‘𝐼)))
113		simplr 768	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐼 ∈ V)
114	31	adantrr 716	. . . . . . . 8 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥:{𝐼}⟶(Base‘𝐾))
115	114	ffnd 6488	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 Fn {𝐼})
116		fnsnbt 39414	. . . . . . . 8 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
117	116	biimpd 232	. . . . . . 7 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} → 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
118	113, 115, 117	sylc 65	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩})
119		opeq2 4765	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝐼) → ⟨𝐼, 𝑦⟩ = ⟨𝐼, (𝑥‘𝐼)⟩)
120	119	sneqd 4537	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝐼) → {⟨𝐼, 𝑦⟩} = {⟨𝐼, (𝑥‘𝐼)⟩})
121	120	eqeq2d 2809	. . . . . 6 ⊢ (𝑦 = (𝑥‘𝐼) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
122	118, 121	syl5ibrcom 250	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑦 = (𝑥‘𝐼) → 𝑥 = {⟨𝐼, 𝑦⟩}))
123	112, 122	impbid 215	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑦 = (𝑥‘𝐼)))
124	36, 35, 104, 123	f1o2d 7379	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)–1-1-onto→(Base‘𝐾))
125	20	a1i 11	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
126	125	f1oeq3d 6587	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (𝐹:(Base‘𝑊)–1-1-onto→(Base‘𝐾) ↔ 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾))))
127	124, 126	mpbid 235	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾)))
128		eqid 2798	. . 3 ⊢ (Base‘(ringLMod‘𝐾)) = (Base‘(ringLMod‘𝐾))
129	1, 128	islmim 19827	. 2 ⊢ (𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)) ↔ (𝐹 ∈ (𝑊 LMHom (ringLMod‘𝐾)) ∧ 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾))))
130	96, 127, 129	sylanbrc 586	1 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531   ↦ cmpt 5110   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Fincfn 8492  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  Scalarcsca 16560   ·_𝑠 cvsca 16561  Grpcgrp 18095  Ringcrg 19290  LModclmod 19627   LMHom clmhm 19784   LMIso clmim 19785  ringLModcrglmod 19934   freeLMod cfrlm 20435

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-prds 16713  df-pws 16715  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-ghm 18348  df-mgp 19233  df-ur 19245  df-ring 19292  df-subrg 19526  df-lmod 19629  df-lss 19697  df-lmhm 19787  df-lmim 19788  df-sra 19937  df-rgmod 19938  df-dsmm 20421  df-frlm 20436

This theorem is referenced by: (None)