Theorem frlmsnic 43744

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 2763	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
3		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘(ringLMod‘𝐾)) = ( ·𝑠 ‘(ringLMod‘𝐾))
4		eqid 2763	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2763	. . 3 ⊢ (Scalar‘(ringLMod‘𝐾)) = (Scalar‘(ringLMod‘𝐾))
6		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
7		snex 5400	. . . . 5 ⊢ {𝐼} ∈ V
8		frlmsnic.w	. . . . . 6 ⊢ 𝑊 = (𝐾 freeLMod {𝐼})
9	8	frlmlmod 22067	. . . . 5 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝑊 ∈ LMod)
10	7, 9	mpan2 704	. . . 4 ⊢ (𝐾 ∈ Ring → 𝑊 ∈ LMod)
11	10	adantr 486	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
12		rlmlmod 21490	. . . 4 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ LMod)
13	12	adantr 486	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ LMod)
14		rlmsca 21485	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
15	14	adantr 486	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘(ringLMod‘𝐾)))
16	8	frlmsca 22071	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ V) → 𝐾 = (Scalar‘𝑊))
17	7, 16	mpan2 704	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑊))
18	17	adantr 486	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐾 = (Scalar‘𝑊))
19	15, 18	eqtr3d 2800	. . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘(ringLMod‘𝐾)) = (Scalar‘𝑊))
20		rlmbas 21480	. . . 4 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾))
21		eqid 2763	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
22		rlmplusg 21481	. . . 4 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾))
23		lmodgrp 21154	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
24	11, 23	syl 18	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ Grp)
25		lmodgrp 21154	. . . . . 6 ⊢ ((ringLMod‘𝐾) ∈ LMod → (ringLMod‘𝐾) ∈ Grp)
26	12, 25	syl 18	. . . . 5 ⊢ (𝐾 ∈ Ring → (ringLMod‘𝐾) ∈ Grp)
27	26	adantr 486	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (ringLMod‘𝐾) ∈ Grp)
28		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
29	8, 28, 1	frlmbasf 22078	. . . . . . . 8 ⊢ (({𝐼} ∈ V ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
30	7, 29	mpan 703	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑊) → 𝑥:{𝐼}⟶(Base‘𝐾))
31	30	adantl 487	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥:{𝐼}⟶(Base‘𝐾))
32		snidg 4620	. . . . . . . 8 ⊢ (𝐼 ∈ V → 𝐼 ∈ {𝐼})
33	32	adantl 487	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐼 ∈ {𝐼})
34	33	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝐼 ∈ {𝐼})
35	31, 34	ffvelcdmd 7093	. . . . 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 7122	. . . 4 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)⟶(Base‘𝐾))
38		simpll 779	. . . . . 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 783	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41		simprr 785	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
42	33	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
43		eqid 2763	. . . . . 6 ⊢ (+g‘𝐾) = (+g‘𝐾)
44	8, 1, 38, 39, 40, 41, 42, 43, 21	frlmvplusgvalc 22085	. . . . 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 486	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
46	1, 21	lmodvacl 21162	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
47	45, 40, 41, 46	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
48		fveq1 6892	. . . . . . 7 ⊢ (𝑡 = (𝑥(+g‘𝑊)𝑦) → (𝑡‘𝐼) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
49		fveq1 6892	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥‘𝐼) = (𝑡‘𝐼))
50	49	cbvmptv 5211	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
51	36, 50	eqtri 2786	. . . . . . 7 ⊢ 𝐹 = (𝑡 ∈ (Base‘𝑊) ↦ (𝑡‘𝐼))
52		fvexd 6908	. . . . . . 7 ⊢ (𝑡 ∈ (Base‘𝑊) → (𝑡‘𝐼) ∈ V)
53	48, 51, 52	fvmpt3 7006	. . . . . 6 ⊢ ((𝑥(+g‘𝑊)𝑦) ∈ (Base‘𝑊) → (𝐹‘(𝑥(+g‘𝑊)𝑦)) = ((𝑥(+g‘𝑊)𝑦)‘𝐼))
54	47, 53	syl 18	. . . . 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 6908	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥‘𝐼) ∈ V)
57	55, 56	fvmpt2d 7015	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝐹‘𝑥) = (𝑥‘𝐼))
58	40, 57	mpdan 700	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑥) = (𝑥‘𝐼))
59		fveq1 6892	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝐼) = (𝑦‘𝐼))
60		fvexd 6908	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑊) → (𝑥‘𝐼) ∈ V)
61	59, 36, 60	fvmpt3 7006	. . . . . . 7 ⊢ (𝑦 ∈ (Base‘𝑊) → (𝐹‘𝑦) = (𝑦‘𝐼))
62	41, 61	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑦) = (𝑦‘𝐼))
63	58, 62	oveq12d 7445	. . . . 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 2808	. . . 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 19451	. . 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 2769	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Scalar‘𝑊) = 𝐾)
68	67	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (Base‘(Scalar‘𝑊)) = (Base‘𝐾))
69	68	eleq2d 2849	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ↔ 𝑥 ∈ (Base‘𝐾)))
70	69	biimpa 482	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
71	70	adantrr 730	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝐾))
72		simprr 785	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
73	33	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝐼 ∈ {𝐼})
74		eqid 2763	. . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾)
75	8, 1, 28, 66, 71, 72, 73, 2, 74	frlmvscaval 22086	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) = (𝑥(.r‘𝐾)(𝑦‘𝐼)))
76		rlmvsca 21487	. . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾))
77	76	oveqi 7440	. . . . 5 ⊢ (𝑥(.r‘𝐾)(𝑦‘𝐼)) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))(𝑦‘𝐼))
78	75, 77	eqtrdi 2814	. . . 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 6892	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥‘𝐼) = (𝑢‘𝐼))
80	79	cbvmptv 5211	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼)) = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
81	36, 80	eqtri 2786	. . . . 5 ⊢ 𝐹 = (𝑢 ∈ (Base‘𝑊) ↦ (𝑢‘𝐼))
82		fveq1 6892	. . . . 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 605	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝑊 ∈ LMod)
85	84	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
86		simprl 783	. . . . . 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 21166	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
88		fvexd 6908	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼) ∈ V)
89	81, 82, 87, 88	fvmptd3 7025	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = ((𝑥( ·𝑠 ‘𝑊)𝑦)‘𝐼))
90		fvex 6906	. . . . . . 7 ⊢ (𝑥‘𝐼) ∈ V
91	59, 36, 90	fvmpt3i 7007	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑊) → (𝐹‘𝑦) = (𝑦‘𝐼))
92	72, 91	syl 18	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐹‘𝑦) = (𝑦‘𝐼))
93	92	oveq2d 7443	. . . 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 2808	. . 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 21326	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMHom (ringLMod‘𝐾)))
96		simplr 781	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐼 ∈ V)
97		simpr 490	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ (Base‘𝐾))
98	96, 97	fsnd 6877	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾))
99		simpll 779	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝐾 ∈ Ring)
100		snfi 9082	. . . . . 6 ⊢ {𝐼} ∈ Fin
101	8, 28, 1	frlmfielbas 43707	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ {𝐼} ∈ Fin) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
102	99, 100, 101	sylancl 598	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → ({⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊) ↔ {⟨𝐼, 𝑦⟩}:{𝐼}⟶(Base‘𝐾)))
103	98, 102	mpbird 260	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ 𝑦 ∈ (Base‘𝐾)) → {⟨𝐼, 𝑦⟩} ∈ (Base‘𝑊))
104		fveq1 6892	. . . . . . . 8 ⊢ (𝑥 = {⟨𝐼, 𝑦⟩} → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
105	104	adantl 487	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → (𝑥‘𝐼) = ({⟨𝐼, 𝑦⟩}‘𝐼))
106		simpllr 788	. . . . . . . 8 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝐼 ∈ V)
107		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
108		fvsng 7192	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑦 ∈ V) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
109	106, 107, 108	sylancl 598	. . . . . . 7 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → ({⟨𝐼, 𝑦⟩}‘𝐼) = 𝑦)
110	105, 109	eqtr2d 2799	. . . . . 6 ⊢ ((((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) ∧ 𝑥 = {⟨𝐼, 𝑦⟩}) → 𝑦 = (𝑥‘𝐼))
111	110	ex 418	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} → 𝑦 = (𝑥‘𝐼)))
112		simplr 781	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝐼 ∈ V)
113	31	adantrr 730	. . . . . . . 8 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥:{𝐼}⟶(Base‘𝐾))
114	113	ffnd 6718	. . . . . . 7 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 Fn {𝐼})
115		fnsnbg 7177	. . . . . . . 8 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
116	115	biimpd 232	. . . . . . 7 ⊢ (𝐼 ∈ V → (𝑥 Fn {𝐼} → 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
117	112, 114, 116	sylc 66	. . . . . 6 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩})
118		opeq2 4843	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝐼) → ⟨𝐼, 𝑦⟩ = ⟨𝐼, (𝑥‘𝐼)⟩)
119	118	sneqd 4595	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝐼) → {⟨𝐼, 𝑦⟩} = {⟨𝐼, (𝑥‘𝐼)⟩})
120	119	eqeq2d 2774	. . . . . 6 ⊢ (𝑦 = (𝑥‘𝐼) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑥 = {⟨𝐼, (𝑥‘𝐼)⟩}))
121	117, 120	syl5ibrcom 250	. . . . 5 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑦 = (𝑥‘𝐼) → 𝑥 = {⟨𝐼, 𝑦⟩}))
122	111, 121	impbid 215	. . . 4 ⊢ (((𝐾 ∈ Ring ∧ 𝐼 ∈ V) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥 = {⟨𝐼, 𝑦⟩} ↔ 𝑦 = (𝑥‘𝐼)))
123	36, 35, 103, 122	f1o2d 7682	. . 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 235	. 2 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹:(Base‘𝑊)–1-1-onto→(Base‘(ringLMod‘𝐾)))
127		eqid 2763	. . 3 ⊢ (Base‘(ringLMod‘𝐾)) = (Base‘(ringLMod‘𝐾))
128	1, 127	islmim 21349	. 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 595	1 ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453  {csn 4587  ⟨cop 4837   ↦ cmpt 5188   Fn wfn 6541  ⟶wf 6542  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7427  Fincfn 8984  Basecbs 17399  +_gcplusg 17440  ._rcmulr 17441  Scalarcsca 17443   ·_𝑠 cvsca 17444  Grpcgrp 19156  Ringcrg 20471  LModclmod 21147   LMHom clmhm 21306   LMIso clmim 21307  ringLModcrglmod 21459   freeLMod cfrlm 22064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-of 7700  df-om 7885  df-1st 8008  df-2nd 8009  df-supp 8188  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-er 8728  df-map 8860  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9365  df-sup 9444  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-3 12416  df-4 12417  df-5 12418  df-6 12419  df-7 12420  df-8 12421  df-9 12422  df-n0 12617  df-z 12704  df-dec 12825  df-uz 12976  df-fz 13650  df-struct 17337  df-sets 17354  df-slot 17372  df-ndx 17384  df-base 17400  df-ress 17421  df-plusg 17453  df-mulr 17454  df-sca 17456  df-vsca 17457  df-ip 17458  df-tset 17459  df-ple 17460  df-ds 17462  df-hom 17464  df-cco 17465  df-0g 17624  df-prds 17630  df-pws 17632  df-mgm 18828  df-sgrp 18920  df-mnd 18936  df-grp 19159  df-minusg 19160  df-sbg 19161  df-subg 19345  df-ghm 19440  df-cmn 20008  df-abl 20009  df-mgp 20373  df-rng 20387  df-ur 20420  df-ring 20473  df-subrg 20834  df-lmod 21149  df-lss 21219  df-lmhm 21309  df-lmim 21310  df-sra 21460  df-rgmod 21461  df-dsmm 22050  df-frlm 22065

This theorem is used by: (None)