Theorem frlmlbs 21928

Description: The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)

Hypotheses

Ref	Expression
frlmlbs.f	⊢ 𝐹 = (𝑅 freeLMod 𝐼)
frlmlbs.u	⊢ 𝑈 = (𝑅 unitVec 𝐼)
frlmlbs.j	⊢ 𝐽 = (LBasis‘𝐹)

Assertion

Ref	Expression
frlmlbs	⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)

Proof of Theorem frlmlbs

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmlbs.u	. . . 4 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
2		frlmlbs.f	. . . 4 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
3		eqid 2763	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	1, 2, 3	uvcff 21922	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶(Base‘𝐹))
5	4	frnd 6716	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ⊆ (Base‘𝐹))
6		suppssdm 8174	. . . . . 6 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
7		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	2, 7, 3	frlmbasf 21891	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
9	8	adantll 726	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
10	6, 9	fssdm 6727	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → (𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
11	10	ralrimiva 3157	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
12		rabid2 3449	. . . 4 ⊢ ((Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼} ↔ ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
13	11, 12	sylibr 237	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
14		ssid 3960	. . . 4 ⊢ 𝐼 ⊆ 𝐼
15		eqid 2763	. . . . 5 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
16		eqid 2763	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
17		eqid 2763	. . . . 5 ⊢ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼}
18	2, 1, 15, 3, 16, 17	frlmsslsp 21927	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐼) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
19	14, 18	mp3an3 1479	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
20		ffn 6707	. . . . 5 ⊢ (𝑈:𝐼⟶(Base‘𝐹) → 𝑈 Fn 𝐼)
21		fnima 6667	. . . . 5 ⊢ (𝑈 Fn 𝐼 → (𝑈 “ 𝐼) = ran 𝑈)
22	4, 20, 21	3syl 19	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝑈 “ 𝐼) = ran 𝑈)
23	22	fveq2d 6887	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = ((LSpan‘𝐹)‘ran 𝑈))
24	13, 19, 23	3eqtr2rd 2805	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹))
25		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘𝐹) = ( ·𝑠 ‘𝐹)
26		eqid 2763	. . . . . 6 ⊢ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})}
27		simpll 778	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ Ring)
28		simplr 780	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝐼 ∈ 𝑉)
29		difssd 4092	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ {𝑐}) ⊆ 𝐼)
30		vsnid 4630	. . . . . . 7 ⊢ 𝑐 ∈ {𝑐}
31		snssi 4752	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐼 → {𝑐} ⊆ 𝐼)
32	31	ad2antrl 740	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {𝑐} ⊆ 𝐼)
33		dfss4 4223	. . . . . . . 8 ⊢ ({𝑐} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
34	32, 33	sylib 221	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
35	30, 34	eleqtrrid 2870	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ (𝐼 ∖ (𝐼 ∖ {𝑐})))
36	2	frlmsca 21884	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
37	36	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
38	36	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g‘𝑅) = (0g‘(Scalar‘𝐹)))
39	38	sneqd 4602	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → {(0g‘𝑅)} = {(0g‘(Scalar‘𝐹))})
40	37, 39	difeq12d 4083	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))
41	40	eleq2d 2849	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ↔ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})))
42	41	biimpar 482	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
43	42	adantrl 728	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
44	2, 1, 3, 7, 25, 16, 26, 27, 28, 29, 35, 43	frlmssuvc2 21926	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})})
45	16, 7	ringelnzr 20608	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → 𝑅 ∈ NzRing)
46	27, 43, 45	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ NzRing)
47	1, 2, 3	uvcf1 21923	. . . . . . . . . 10 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼–1-1→(Base‘𝐹))
48	46, 28, 47	syl2anc 595	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑈:𝐼–1-1→(Base‘𝐹))
49		df-f1 6543	. . . . . . . . . 10 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) ↔ (𝑈:𝐼⟶(Base‘𝐹) ∧ Fun ◡𝑈))
50	49	simprbi 502	. . . . . . . . 9 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) → Fun ◡𝑈)
51		imadif 6622	. . . . . . . . 9 ⊢ (Fun ◡𝑈 → (𝑈 “ (𝐼 ∖ {𝑐})) = ((𝑈 “ 𝐼) ∖ (𝑈 “ {𝑐})))
52	48, 50, 51	3syl 19	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ (𝐼 ∖ {𝑐})) = ((𝑈 “ 𝐼) ∖ (𝑈 “ {𝑐})))
53		f1fn 6777	. . . . . . . . . 10 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) → 𝑈 Fn 𝐼)
54	48, 53, 21	3syl 19	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ 𝐼) = ran 𝑈)
55	48, 53	syl 18	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑈 Fn 𝐼)
56		simprl 782	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ 𝐼)
57		fnsnfv 6962	. . . . . . . . . . 11 ⊢ ((𝑈 Fn 𝐼 ∧ 𝑐 ∈ 𝐼) → {(𝑈‘𝑐)} = (𝑈 “ {𝑐}))
58	55, 56, 57	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {(𝑈‘𝑐)} = (𝑈 “ {𝑐}))
59	58	eqcomd 2769	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ {𝑐}) = {(𝑈‘𝑐)})
60	54, 59	difeq12d 4083	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((𝑈 “ 𝐼) ∖ (𝑈 “ {𝑐})) = (ran 𝑈 ∖ {(𝑈‘𝑐)}))
61	52, 60	eqtr2d 2799	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (ran 𝑈 ∖ {(𝑈‘𝑐)}) = (𝑈 “ (𝐼 ∖ {𝑐})))
62	61	fveq2d 6887	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})) = ((LSpan‘𝐹)‘(𝑈 “ (𝐼 ∖ {𝑐}))))
63	2, 1, 15, 3, 16, 26	frlmsslsp 21927	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ (𝐼 ∖ {𝑐}) ⊆ 𝐼) → ((LSpan‘𝐹)‘(𝑈 “ (𝐼 ∖ {𝑐}))) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})})
64	27, 28, 29, 63	syl3anc 1398	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((LSpan‘𝐹)‘(𝑈 “ (𝐼 ∖ {𝑐}))) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})})
65	62, 64	eqtrd 2798	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})})
66	44, 65	neleqtrrd 2886	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
67	66	ralrimivva 3208	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑐 ∈ 𝐼 ∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
68		oveq2 7420	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → (𝑏( ·𝑠 ‘𝐹)𝑎) = (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)))
69		sneq 4600	. . . . . . . . . 10 ⊢ (𝑎 = (𝑈‘𝑐) → {𝑎} = {(𝑈‘𝑐)})
70	69	difeq2d 4082	. . . . . . . . 9 ⊢ (𝑎 = (𝑈‘𝑐) → (ran 𝑈 ∖ {𝑎}) = (ran 𝑈 ∖ {(𝑈‘𝑐)}))
71	70	fveq2d 6887	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) = ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
72	68, 71	eleq12d 2857	. . . . . . 7 ⊢ (𝑎 = (𝑈‘𝑐) → ((𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)}))))
73	72	notbid 321	. . . . . 6 ⊢ (𝑎 = (𝑈‘𝑐) → (¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)}))))
74	73	ralbidv 3188	. . . . 5 ⊢ (𝑎 = (𝑈‘𝑐) → (∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)}))))
75	74	ralrn 7085	. . . 4 ⊢ (𝑈 Fn 𝐼 → (∀𝑎 ∈ ran 𝑈∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ∀𝑐 ∈ 𝐼 ∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)}))))
76	4, 20, 75	3syl 19	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (∀𝑎 ∈ ran 𝑈∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ∀𝑐 ∈ 𝐼 ∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)}))))
77	67, 76	mpbird 260	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑎 ∈ ran 𝑈∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})))
78	2	ovexi 7446	. . 3 ⊢ 𝐹 ∈ V
79		eqid 2763	. . . 4 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
80		eqid 2763	. . . 4 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
81		frlmlbs.j	. . . 4 ⊢ 𝐽 = (LBasis‘𝐹)
82		eqid 2763	. . . 4 ⊢ (0g‘(Scalar‘𝐹)) = (0g‘(Scalar‘𝐹))
83	3, 79, 25, 80, 81, 15, 82	islbs 21178	. . 3 ⊢ (𝐹 ∈ V → (ran 𝑈 ∈ 𝐽 ↔ (ran 𝑈 ⊆ (Base‘𝐹) ∧ ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹) ∧ ∀𝑎 ∈ ran 𝑈∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})))))
84	78, 83	ax-mp 5	. 2 ⊢ (ran 𝑈 ∈ 𝐽 ↔ (ran 𝑈 ⊆ (Base‘𝐹) ∧ ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹) ∧ ∀𝑎 ∈ ran 𝑈∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠 ‘𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎}))))
85	5, 24, 77, 84	syl3anbrc 1362	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  {csn 4590  ^◡ccnv 5662  ran crn 5664   “ cima 5666  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7412   supp csupp 8157  Basecbs 17270  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  Ringcrg 20316  NzRingcnzr 20596  LSpanclspn 21073  LBasisclbs 21176   freeLMod cfrlm 21877   unitVec cuvc 21913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-sup 9403  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17495  df-gsum 17496  df-prds 17501  df-pws 17503  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-grp 19004  df-minusg 19005  df-sbg 19006  df-mulg 19135  df-subg 19190  df-ghm 19285  df-cntz 19388  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-nzr 20597  df-subrg 20656  df-lmod 20964  df-lss 21034  df-lsp 21074  df-lmhm 21124  df-lbs 21177  df-sra 21275  df-rgmod 21276  df-dsmm 21863  df-frlm 21878  df-uvc 21914

This theorem is referenced by:  frlmup3  21931  frlmup4  21932  lmisfree  21973  frlmisfrlm  21979  frlmdim  33979  lindsdom  38243  aacllem  50578