Theorem frlmsnic 43002

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 2737	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
3		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘(ringLMod‘𝐾)) = ( ·𝑠 ‘(ringLMod‘𝐾))
4		eqid 2737	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2737	. . 3 ⊢ (Scalar‘(ringLMod‘𝐾)) = (Scalar‘(ringLMod‘𝐾))
6		eqid 2737	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7		snex 5377	. . . . 5 ⊢ {𝐼} ∈ V
8		frlmsnic.w	. . . . . 6 ⊢ 𝑊 = (𝐾 freeLMod {𝐼})
9	8	frlmlmod 21742	. . . . 5 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝑊 ∈ LMod)
10	7, 9	mpan2 692	. . . 4 ⊢ (𝐾 ∈ Ring → 𝑊 ∈ LMod)
11	10	adantr 480	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
12		rlmlmod 21193	. . . 4 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ LMod)
13	12	adantr 480	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ LMod)
14		rlmsca 21188	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
15	14	adantr 480	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
16	8	frlmsca 21746	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝐾 = (Scalar‘𝑊))
17	7, 16	mpan2 692	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑊))
18	17	adantr 480	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘𝑊))
19	15, 18	eqtr3d 2774	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘(ringLMod‘𝐾)) = (Scalar‘𝑊))
20		rlmbas 21183	. . . 4 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾))
21		eqid 2737	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
22		rlmplusg 21184	. . . 4 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾))
23		lmodgrp 20856	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
24	11, 23	syl 17	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ Grp)
25		lmodgrp 20856	. . . . . 6 ⊢ ((ringLMod‘𝐾) ∈ LMod → (ringLMod‘𝐾) ∈ Grp)
26	12, 25	syl 17	. . . . 5 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ Grp)
27	26	adantr 480	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ Grp)
28		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
29	8, 28, 1	frlmbasf 21753	. . . . . . . 8 ⊢ (({𝐼} ∈ V ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
30	7, 29	mpan 691	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑊) → 𝑥:{𝐼}⟶(Base‘𝐾))
31	30	adantl 481	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
32		snidg 4605	. . . . . . . 8 ⊢ (𝐼 ∈ V → 𝐼 ∈ {𝐼})
33	32	adantl 481	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐼 ∈ {𝐼})
34	33	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝐼 ∈ {𝐼})
35	31, 34	ffvelcdmd 7032	. . . . 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 7061	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)⟶(Base‘𝐾))
38		simpll 767	. . . . . 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 771	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41		simprr 773	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
42	33	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
43		eqid 2737	. . . . . 6 ⊢ (+g‘𝐾) = (+g‘𝐾)
44	8, 1, 38, 39, 40, 41, 42, 43, 21	frlmvplusgvalc 21760	. . . . 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 480	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
46	1, 21	lmodvacl 20864	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
47	45, 40, 41, 46	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
48		fveq1 6834	. . . . . . 7 ⊢ (𝑡 = (𝑥(+g‘𝑊)𝑦) → (𝑡‘𝐼) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
49		fveq1 6834	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥‘𝐼) = (𝑡‘𝐼))
50	49	cbvmptv 5190	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
51	36, 50	eqtri 2760	. . . . . . 7 ⊢ 𝐹 = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
52		fvexd 6850	. . . . . . 7 ⊢ (𝑡 ∈ (Base‘𝑊) → (𝑡‘𝐼) ∈ V)
53	48, 51, 52	fvmpt3 6947	. . . . . 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 6850	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥‘𝐼) ∈ V)
57	55, 56	fvmpt2d 6956	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝐹‘𝑥) = (𝑥‘𝐼))
58	40, 57	mpdan 688	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑥) = (𝑥‘𝐼))
59		fveq1 6834	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝐼) = (𝑦‘𝐼))
60		fvexd 6850	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) → (𝑥‘𝐼) ∈ V)
61	59, 36, 60	fvmpt3 6947	. . . . . . 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 7379	. . . . 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 2782	. . . 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 19194	. . 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 2743	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘𝑊) = 𝐾)
68	67	fveq2d 6839	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Base‘(Scalar‘𝑊)) = (Base‘𝐾))
69	68	eleq2d 2823	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↔ 𝑥 ∈ (Base‘𝐾)))
70	69	biimpa 476	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
71	70	adantrr 718	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
72		simprr 773	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
73	33	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
74		eqid 2737	. . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾)
75	8, 1, 28, 66, 71, 72, 73, 2, 74	frlmvscaval 21761	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) = (𝑥(.r‘𝐾)(𝑦‘𝐼)))
76		rlmvsca 21190	. . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾))
77	76	oveqi 7374	. . . . 5 ⊢ (𝑥(.r‘𝐾)(𝑦‘𝐼)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼))
78	75, 77	eqtrdi 2788	. . . 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 6834	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥‘𝐼) = (𝑢‘𝐼))
80	79	cbvmptv 5190	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
81	36, 80	eqtri 2760	. . . . 5 ⊢ 𝐹 = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
82		fveq1 6834	. . . . 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 594	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
85	84	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
86		simprl 771	. . . . . 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 20868	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
88		fvexd 6850	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) ∈ V)
89	81, 82, 87, 88	fvmptd3 6966	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼))
90		fvex 6848	. . . . . . 7 ⊢ (𝑥‘𝐼) ∈ V
91	59, 36, 90	fvmpt3i 6948	. . . . . 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 7377	. . . 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 2782	. . 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 21029	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMHom (ringLMod‘𝐾)))
96		simplr 769	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐼 ∈ V)
97		simpr 484	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
98	96, 97	fsnd 6819	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾))
99		simpll 767	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐾 ∈ Ring)
100		snfi 8984	. . . . . 6 ⊢ {𝐼} ∈ Fin
101	8, 28, 1	frlmfielbas 42962	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ Fin) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
102	99, 100, 101	sylancl 587	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
103	98, 102	mpbird 257	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊))
104		fveq1 6834	. . . . . . . 8 ⊢ (𝑥 = {⟨𝐼, 𝑦⟩} → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
105	104	adantl 481	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
106		simpllr 776	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝐼 ∈ V)
107		vex 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
108		fvsng 7129	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑦 ∈ V) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
109	106, 107, 108	sylancl 587	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
110	105, 109	eqtr2d 2773	. . . . . 6 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝑦 = (𝑥‘𝐼))
111	110	ex 412	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} → 𝑦 = (𝑥‘𝐼)))
112		simplr 769	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐼 ∈ V)
113	31	adantrr 718	. . . . . . . 8 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥:{𝐼}⟶(Base‘𝐾))
114	113	ffnd 6664	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 Fn {𝐼})
115		fnsnbg 7113	. . . . . . . 8 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
116	115	biimpd 229	. . . . . . 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 4818	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝐼) → ⟨𝐼, 𝑦⟩ = ⟨𝐼, (𝑥‘𝐼)⟩)
119	118	sneqd 4580	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝐼) → {⟨𝐼, 𝑦⟩} = {⟨𝐼, (𝑥‘𝐼)⟩})
120	119	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 = (𝑥‘𝐼) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
121	117, 120	syl5ibrcom 247	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑦 = (𝑥‘𝐼) → 𝑥 = {⟨𝐼, 𝑦⟩}))
122	111, 121	impbid 212	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑦 = (𝑥‘𝐼)))
123	36, 35, 103, 122	f1o2d 7615	. . 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 6772	. . 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 232	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾)))
127		eqid 2737	. . 3 ⊢ (Base‘(ringLMod‘𝐾)) = (Base‘(ringLMod‘𝐾))
128	1, 127	islmim 21052	. 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 584	1 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {csn 4568  ⟨cop 4574   ↦ cmpt 5167   Fn wfn 6488  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  Fincfn 8887  Basecbs 17173  +_gcplusg 17214  ._rcmulr 17215  Scalarcsca 17217   ·_𝑠 cvsca 17218  Grpcgrp 18903  Ringcrg 20208  LModclmod 20849   LMHom clmhm 21009   LMIso clmim 21010  ringLModcrglmod 21162   freeLMod cfrlm 21739

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-0g 17398  df-prds 17404  df-pws 17406  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906  df-minusg 18907  df-sbg 18908  df-subg 19093  df-ghm 19182  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-subrg 20541  df-lmod 20851  df-lss 20921  df-lmhm 21012  df-lmim 21013  df-sra 21163  df-rgmod 21164  df-dsmm 21725  df-frlm 21740

This theorem is referenced by: (None)