Theorem frlmlbs 20611

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 2793	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	1, 2, 3	uvcff 20605	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶(Base‘𝐹))
5	4	frnd 6381	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ⊆ (Base‘𝐹))
6		suppssdm 7685	. . . . . 6 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
7		eqid 2793	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	2, 7, 3	frlmbasf 20574	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
9	8	adantll 710	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
10	6, 9	fssdm 6390	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → (𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
11	10	ralrimiva 3147	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
12		rabid2 3337	. . . 4 ⊢ ((Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼} ↔ ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
13	11, 12	sylibr 235	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
14		ssid 3905	. . . 4 ⊢ 𝐼 ⊆ 𝐼
15		eqid 2793	. . . . 5 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
16		eqid 2793	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
17		eqid 2793	. . . . 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 20610	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐼) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
19	14, 18	mp3an3 1440	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
20		ffn 6374	. . . . 5 ⊢ (𝑈:𝐼⟶(Base‘𝐹) → 𝑈 Fn 𝐼)
21		fnima 6338	. . . . 5 ⊢ (𝑈 Fn 𝐼 → (𝑈 “ 𝐼) = ran 𝑈)
22	4, 20, 21	3syl 18	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝑈 “ 𝐼) = ran 𝑈)
23	22	fveq2d 6534	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = ((LSpan‘𝐹)‘ran 𝑈))
24	13, 19, 23	3eqtr2rd 2836	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹))
25		eqid 2793	. . . . . 6 ⊢ ( ·𝑠 ‘𝐹) = ( ·𝑠 ‘𝐹)
26		eqid 2793	. . . . . 6 ⊢ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})}
27		simpll 763	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ Ring)
28		simplr 765	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝐼 ∈ 𝑉)
29		difssd 4025	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ {𝑐}) ⊆ 𝐼)
30		vsnid 4501	. . . . . . 7 ⊢ 𝑐 ∈ {𝑐}
31		snssi 4642	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐼 → {𝑐} ⊆ 𝐼)
32	31	ad2antrl 724	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {𝑐} ⊆ 𝐼)
33		dfss4 4150	. . . . . . . 8 ⊢ ({𝑐} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
34	32, 33	sylib 219	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
35	30, 34	syl5eleqr 2888	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ (𝐼 ∖ (𝐼 ∖ {𝑐})))
36	2	frlmsca 20567	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
37	36	fveq2d 6534	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
38	36	fveq2d 6534	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g‘𝑅) = (0g‘(Scalar‘𝐹)))
39	38	sneqd 4478	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → {(0g‘𝑅)} = {(0g‘(Scalar‘𝐹))})
40	37, 39	difeq12d 4016	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))
41	40	eleq2d 2866	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ↔ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})))
42	41	biimpar 478	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
43	42	adantrl 712	. . . . . 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 20609	. . . . 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 19716	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → 𝑅 ∈ NzRing)
46	27, 43, 45	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ NzRing)
47	1, 2, 3	uvcf1 20606	. . . . . . . . . 10 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼–1-1→(Base‘𝐹))
48	46, 28, 47	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑈:𝐼–1-1→(Base‘𝐹))
49		df-f1 6222	. . . . . . . . . 10 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) ↔ (𝑈:𝐼⟶(Base‘𝐹) ∧ Fun ◡𝑈))
50	49	simprbi 497	. . . . . . . . 9 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) → Fun ◡𝑈)
51		imadif 6300	. . . . . . . . 9 ⊢ (Fun ◡𝑈 → (𝑈 “ (𝐼 ∖ {𝑐})) = ((𝑈 “ 𝐼) ∖ (𝑈 “ {𝑐})))
52	48, 50, 51	3syl 18	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ (𝐼 ∖ {𝑐})) = ((𝑈 “ 𝐼) ∖ (𝑈 “ {𝑐})))
53		f1fn 6436	. . . . . . . . . 10 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) → 𝑈 Fn 𝐼)
54	48, 53, 21	3syl 18	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ 𝐼) = ran 𝑈)
55	48, 53	syl 17	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑈 Fn 𝐼)
56		simprl 767	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ 𝐼)
57		fnsnfv 6602	. . . . . . . . . . 11 ⊢ ((𝑈 Fn 𝐼 ∧ 𝑐 ∈ 𝐼) → {(𝑈‘𝑐)} = (𝑈 “ {𝑐}))
58	55, 56, 57	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {(𝑈‘𝑐)} = (𝑈 “ {𝑐}))
59	58	eqcomd 2799	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ {𝑐}) = {(𝑈‘𝑐)})
60	54, 59	difeq12d 4016	. . . . . . . 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 2830	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (ran 𝑈 ∖ {(𝑈‘𝑐)}) = (𝑈 “ (𝐼 ∖ {𝑐})))
62	61	fveq2d 6534	. . . . . 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 20610	. . . . . . 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 1362	. . . . . 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 2829	. . . . 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 2903	. . . 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 3156	. . 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 7015	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → (𝑏( ·𝑠 ‘𝐹)𝑎) = (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)))
69		sneq 4476	. . . . . . . . . 10 ⊢ (𝑎 = (𝑈‘𝑐) → {𝑎} = {(𝑈‘𝑐)})
70	69	difeq2d 4015	. . . . . . . . 9 ⊢ (𝑎 = (𝑈‘𝑐) → (ran 𝑈 ∖ {𝑎}) = (ran 𝑈 ∖ {(𝑈‘𝑐)}))
71	70	fveq2d 6534	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) = ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
72	68, 71	eleq12d 2875	. . . . . . 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 319	. . . . . 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 3162	. . . . 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 6710	. . . 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 18	. . 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 258	. 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 7040	. . 3 ⊢ 𝐹 ∈ V
79		eqid 2793	. . . 4 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
80		eqid 2793	. . . 4 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
81		frlmlbs.j	. . . 4 ⊢ 𝐽 = (LBasis‘𝐹)
82		eqid 2793	. . . 4 ⊢ (0g‘(Scalar‘𝐹)) = (0g‘(Scalar‘𝐹))
83	3, 79, 25, 80, 81, 15, 82	islbs 19526	. . 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 1334	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  ∀wral 3103  {crab 3107  Vcvv 3432   ∖ cdif 3851   ⊆ wss 3854  {csn 4466  ^◡ccnv 5434  ran crn 5436   “ cima 5438  Fun wfun 6211   Fn wfn 6212  ⟶wf 6213  –1-1→wf1 6214  ‘cfv 6217  (class class class)co 7007   supp csupp 7672  Basecbs 16300  Scalarcsca 16385   ·_𝑠 cvsca 16386  0_gc0g 16530  Ringcrg 18975  LSpanclspn 19421  LBasisclbs 19524  NzRingcnzr 19707   freeLMod cfrlm 20560   unitVec cuvc 20596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-iin 4822  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-of 7258  df-om 7428  df-1st 7536  df-2nd 7537  df-supp 7673  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-map 8249  df-ixp 8301  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-fsupp 8670  df-sup 8742  df-oi 8810  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-z 11819  df-dec 11937  df-uz 12083  df-fz 12732  df-fzo 12873  df-seq 13208  df-hash 13529  df-struct 16302  df-ndx 16303  df-slot 16304  df-base 16306  df-sets 16307  df-ress 16308  df-plusg 16395  df-mulr 16396  df-sca 16398  df-vsca 16399  df-ip 16400  df-tset 16401  df-ple 16402  df-ds 16404  df-hom 16406  df-cco 16407  df-0g 16532  df-gsum 16533  df-prds 16538  df-pws 16540  df-mre 16674  df-mrc 16675  df-acs 16677  df-mgm 17669  df-sgrp 17711  df-mnd 17722  df-mhm 17762  df-submnd 17763  df-grp 17852  df-minusg 17853  df-sbg 17854  df-mulg 17970  df-subg 18018  df-ghm 18085  df-cntz 18176  df-cmn 18623  df-abl 18624  df-mgp 18918  df-ur 18930  df-ring 18977  df-subrg 19211  df-lmod 19314  df-lss 19382  df-lsp 19422  df-lmhm 19472  df-lbs 19525  df-sra 19622  df-rgmod 19623  df-nzr 19708  df-dsmm 20546  df-frlm 20561  df-uvc 20597

This theorem is referenced by:  frlmup3  20614  frlmup4  20615  lmisfree  20656  frlmisfrlm  20662  frlmdim  30568  lindsdom  34363  aacllem  44336