Theorem frlmsnic 41798

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 2725	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2725	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
3		eqid 2725	. . 3 ⊢ ( ·𝑠 ‘(ringLMod‘𝐾)) = ( ·𝑠 ‘(ringLMod‘𝐾))
4		eqid 2725	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2725	. . 3 ⊢ (Scalar‘(ringLMod‘𝐾)) = (Scalar‘(ringLMod‘𝐾))
6		eqid 2725	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7		snex 5425	. . . . 5 ⊢ {𝐼} ∈ V
8		frlmsnic.w	. . . . . 6 ⊢ 𝑊 = (𝐾 freeLMod {𝐼})
9	8	frlmlmod 21685	. . . . 5 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝑊 ∈ LMod)
10	7, 9	mpan2 689	. . . 4 ⊢ (𝐾 ∈ Ring → 𝑊 ∈ LMod)
11	10	adantr 479	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
12		rlmlmod 21098	. . . 4 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ LMod)
13	12	adantr 479	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ LMod)
14		rlmsca 21093	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
15	14	adantr 479	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
16	8	frlmsca 21689	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝐾 = (Scalar‘𝑊))
17	7, 16	mpan2 689	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑊))
18	17	adantr 479	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘𝑊))
19	15, 18	eqtr3d 2767	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘(ringLMod‘𝐾)) = (Scalar‘𝑊))
20		rlmbas 21088	. . . 4 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾))
21		eqid 2725	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
22		rlmplusg 21089	. . . 4 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾))
23		lmodgrp 20752	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
24	11, 23	syl 17	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ Grp)
25		lmodgrp 20752	. . . . . 6 ⊢ ((ringLMod‘𝐾) ∈ LMod → (ringLMod‘𝐾) ∈ Grp)
26	12, 25	syl 17	. . . . 5 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ Grp)
27	26	adantr 479	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ Grp)
28		eqid 2725	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
29	8, 28, 1	frlmbasf 21696	. . . . . . . 8 ⊢ (({𝐼} ∈ V ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
30	7, 29	mpan 688	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑊) → 𝑥:{𝐼}⟶(Base‘𝐾))
31	30	adantl 480	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
32		snidg 4656	. . . . . . . 8 ⊢ (𝐼 ∈ V → 𝐼 ∈ {𝐼})
33	32	adantl 480	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐼 ∈ {𝐼})
34	33	adantr 479	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝐼 ∈ {𝐼})
35	31, 34	ffvelcdmd 7088	. . . . 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 7117	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)⟶(Base‘𝐾))
38		simpll 765	. . . . . 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 769	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41		simprr 771	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
42	33	adantr 479	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
43		eqid 2725	. . . . . 6 ⊢ (+g‘𝐾) = (+g‘𝐾)
44	8, 1, 38, 39, 40, 41, 42, 43, 21	frlmvplusgvalc 21703	. . . . 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 479	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
46	1, 21	lmodvacl 20760	. . . . . . 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 6889	. . . . . . 7 ⊢ (𝑡 = (𝑥(+g‘𝑊)𝑦) → (𝑡‘𝐼) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
49		fveq1 6889	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥‘𝐼) = (𝑡‘𝐼))
50	49	cbvmptv 5254	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
51	36, 50	eqtri 2753	. . . . . . 7 ⊢ 𝐹 = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
52		fvexd 6905	. . . . . . 7 ⊢ (𝑡 ∈ (Base‘𝑊) → (𝑡‘𝐼) ∈ V)
53	48, 51, 52	fvmpt3 7002	. . . . . 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 6905	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥‘𝐼) ∈ V)
57	55, 56	fvmpt2d 7011	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝐹‘𝑥) = (𝑥‘𝐼))
58	40, 57	mpdan 685	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑥) = (𝑥‘𝐼))
59		fveq1 6889	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝐼) = (𝑦‘𝐼))
60		fvexd 6905	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) → (𝑥‘𝐼) ∈ V)
61	59, 36, 60	fvmpt3 7002	. . . . . . 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 7432	. . . . 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 2775	. . . 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 19181	. . 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 2731	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘𝑊) = 𝐾)
68	67	fveq2d 6894	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Base‘(Scalar‘𝑊)) = (Base‘𝐾))
69	68	eleq2d 2811	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↔ 𝑥 ∈ (Base‘𝐾)))
70	69	biimpa 475	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
71	70	adantrr 715	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
72		simprr 771	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
73	33	adantr 479	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
74		eqid 2725	. . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾)
75	8, 1, 28, 66, 71, 72, 73, 2, 74	frlmvscaval 21704	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) = (𝑥(.r‘𝐾)(𝑦‘𝐼)))
76		rlmvsca 21095	. . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾))
77	76	oveqi 7427	. . . . 5 ⊢ (𝑥(.r‘𝐾)(𝑦‘𝐼)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼))
78	75, 77	eqtrdi 2781	. . . 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 6889	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥‘𝐼) = (𝑢‘𝐼))
80	79	cbvmptv 5254	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
81	36, 80	eqtri 2753	. . . . 5 ⊢ 𝐹 = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
82		fveq1 6889	. . . . 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 591	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
85	84	adantr 479	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
86		simprl 769	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
87	1, 4, 2, 6, 85, 86, 72	lmodvscld 20764	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
88		fvexd 6905	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) ∈ V)
89	81, 82, 87, 88	fvmptd3 7021	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼))
90		fvex 6903	. . . . . . 7 ⊢ (𝑥‘𝐼) ∈ V
91	59, 36, 90	fvmpt3i 7003	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑊) → (𝐹‘𝑦) = (𝑦‘𝐼))
92	72, 91	syl 17	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑦) = (𝑦‘𝐼))
93	92	oveq2d 7430	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼)))
94	78, 89, 93	3eqtr4d 2775	. . 3 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝐹‘𝑦)))
95	1, 2, 3, 4, 5, 6, 11, 13, 19, 65, 94	islmhmd 20926	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMHom (ringLMod‘𝐾)))
96		simplr 767	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐼 ∈ V)
97		simpr 483	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
98	96, 97	fsnd 6875	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾))
99		simpll 765	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐾 ∈ Ring)
100		snfi 9065	. . . . . 6 ⊢ {𝐼} ∈ Fin
101	8, 28, 1	frlmfielbas 41768	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ Fin) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
102	99, 100, 101	sylancl 584	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
103	98, 102	mpbird 256	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊))
104		fveq1 6889	. . . . . . . 8 ⊢ (𝑥 = {⟨𝐼, 𝑦⟩} → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
105	104	adantl 480	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
106		simpllr 774	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝐼 ∈ V)
107		vex 3467	. . . . . . . 8 ⊢ 𝑦 ∈ V
108		fvsng 7183	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑦 ∈ V) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
109	106, 107, 108	sylancl 584	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
110	105, 109	eqtr2d 2766	. . . . . 6 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝑦 = (𝑥‘𝐼))
111	110	ex 411	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} → 𝑦 = (𝑥‘𝐼)))
112		simplr 767	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐼 ∈ V)
113	31	adantrr 715	. . . . . . . 8 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥:{𝐼}⟶(Base‘𝐾))
114	113	ffnd 6716	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 Fn {𝐼})
115		fnsnbt 41749	. . . . . . . 8 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
116	115	biimpd 228	. . . . . . 7 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} → 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
117	112, 114, 116	sylc 65	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩})
118		opeq2 4868	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝐼) → ⟨𝐼, 𝑦⟩ = ⟨𝐼, (𝑥‘𝐼)⟩)
119	118	sneqd 4634	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝐼) → {⟨𝐼, 𝑦⟩} = {⟨𝐼, (𝑥‘𝐼)⟩})
120	119	eqeq2d 2736	. . . . . 6 ⊢ (𝑦 = (𝑥‘𝐼) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
121	117, 120	syl5ibrcom 246	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑦 = (𝑥‘𝐼) → 𝑥 = {⟨𝐼, 𝑦⟩}))
122	111, 121	impbid 211	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑦 = (𝑥‘𝐼)))
123	36, 35, 103, 122	f1o2d 7670	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)–1-1-onto→(Base‘𝐾))
124	20	a1i 11	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
125	124	f1oeq3d 6829	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (𝐹:(Base‘𝑊)–1-1-onto→(Base‘𝐾) ↔ 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾))))
126	123, 125	mpbid 231	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾)))
127		eqid 2725	. . 3 ⊢ (Base‘(ringLMod‘𝐾)) = (Base‘(ringLMod‘𝐾))
128	1, 127	islmim 20949	. 2 ⊢ (𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)) ↔ (𝐹 ∈ (𝑊 LMHom (ringLMod‘𝐾)) ∧ 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾))))
129	95, 126, 128	sylanbrc 581	1 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  {csn 4622  ⟨cop 4628   ↦ cmpt 5224   Fn wfn 6536  ⟶wf 6537  –1-1-onto→wf1o 6540  ‘cfv 6541  (class class class)co 7414  Fincfn 8960  Basecbs 17177  +_gcplusg 17230  ._rcmulr 17231  Scalarcsca 17233   ·_𝑠 cvsca 17234  Grpcgrp 18892  Ringcrg 20175  LModclmod 20745   LMHom clmhm 20906   LMIso clmim 20907  ringLModcrglmod 21059   freeLMod cfrlm 21682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-om 7867  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-ixp 8913  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-fsupp 9384  df-sup 9463  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-uz 12851  df-fz 13515  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-mulr 17244  df-sca 17246  df-vsca 17247  df-ip 17248  df-tset 17249  df-ple 17250  df-ds 17252  df-hom 17254  df-cco 17255  df-0g 17420  df-prds 17426  df-pws 17428  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-sbg 18897  df-subg 19080  df-ghm 19170  df-cmn 19739  df-abl 19740  df-mgp 20077  df-rng 20095  df-ur 20124  df-ring 20177  df-subrg 20510  df-lmod 20747  df-lss 20818  df-lmhm 20909  df-lmim 20910  df-sra 21060  df-rgmod 21061  df-dsmm 21668  df-frlm 21683

This theorem is referenced by: (None)