Theorem frlmlbs 21013

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 2739	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	1, 2, 3	uvcff 21007	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶(Base‘𝐹))
5	4	frnd 6617	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ⊆ (Base‘𝐹))
6		suppssdm 8002	. . . . . 6 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
7		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	2, 7, 3	frlmbasf 20976	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
9	8	adantll 711	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
10	6, 9	fssdm 6629	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → (𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
11	10	ralrimiva 3104	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
12		rabid2 3315	. . . 4 ⊢ ((Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼} ↔ ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
13	11, 12	sylibr 233	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
14		ssid 3944	. . . 4 ⊢ 𝐼 ⊆ 𝐼
15		eqid 2739	. . . . 5 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
16		eqid 2739	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
17		eqid 2739	. . . . 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 21012	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐼) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
19	14, 18	mp3an3 1449	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
20		ffn 6609	. . . . 5 ⊢ (𝑈:𝐼⟶(Base‘𝐹) → 𝑈 Fn 𝐼)
21		fnima 6572	. . . . 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 6787	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = ((LSpan‘𝐹)‘ran 𝑈))
24	13, 19, 23	3eqtr2rd 2786	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹))
25		eqid 2739	. . . . . 6 ⊢ ( ·𝑠 ‘𝐹) = ( ·𝑠 ‘𝐹)
26		eqid 2739	. . . . . 6 ⊢ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})}
27		simpll 764	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ Ring)
28		simplr 766	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝐼 ∈ 𝑉)
29		difssd 4068	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ {𝑐}) ⊆ 𝐼)
30		vsnid 4599	. . . . . . 7 ⊢ 𝑐 ∈ {𝑐}
31		snssi 4742	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐼 → {𝑐} ⊆ 𝐼)
32	31	ad2antrl 725	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {𝑐} ⊆ 𝐼)
33		dfss4 4193	. . . . . . . 8 ⊢ ({𝑐} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
34	32, 33	sylib 217	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
35	30, 34	eleqtrrid 2847	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ (𝐼 ∖ (𝐼 ∖ {𝑐})))
36	2	frlmsca 20969	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
37	36	fveq2d 6787	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
38	36	fveq2d 6787	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g‘𝑅) = (0g‘(Scalar‘𝐹)))
39	38	sneqd 4574	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → {(0g‘𝑅)} = {(0g‘(Scalar‘𝐹))})
40	37, 39	difeq12d 4059	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))
41	40	eleq2d 2825	. . . . . . . 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 713	. . . . . 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 21011	. . . . 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 20546	. . . . . . . . . . 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 21008	. . . . . . . . . 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 6442	. . . . . . . . . 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 6525	. . . . . . . . 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 6680	. . . . . . . . . 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 768	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ 𝐼)
57		fnsnfv 6856	. . . . . . . . . . 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 2745	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ {𝑐}) = {(𝑈‘𝑐)})
60	54, 59	difeq12d 4059	. . . . . . . 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 2780	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (ran 𝑈 ∖ {(𝑈‘𝑐)}) = (𝑈 “ (𝐼 ∖ {𝑐})))
62	61	fveq2d 6787	. . . . . 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 21012	. . . . . . 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 1370	. . . . . 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 2779	. . . . 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 2862	. . . 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 3124	. . 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 7292	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → (𝑏( ·𝑠 ‘𝐹)𝑎) = (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)))
69		sneq 4572	. . . . . . . . . 10 ⊢ (𝑎 = (𝑈‘𝑐) → {𝑎} = {(𝑈‘𝑐)})
70	69	difeq2d 4058	. . . . . . . . 9 ⊢ (𝑎 = (𝑈‘𝑐) → (ran 𝑈 ∖ {𝑎}) = (ran 𝑈 ∖ {(𝑈‘𝑐)}))
71	70	fveq2d 6787	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) = ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
72	68, 71	eleq12d 2834	. . . . . . 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 318	. . . . . 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 3113	. . . . 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 6973	. . . 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 256	. 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 7318	. . 3 ⊢ 𝐹 ∈ V
79		eqid 2739	. . . 4 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
80		eqid 2739	. . . 4 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
81		frlmlbs.j	. . . 4 ⊢ 𝐽 = (LBasis‘𝐹)
82		eqid 2739	. . . 4 ⊢ (0g‘(Scalar‘𝐹)) = (0g‘(Scalar‘𝐹))
83	3, 79, 25, 80, 81, 15, 82	islbs 20347	. . 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 1342	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3065  {crab 3069  Vcvv 3433   ∖ cdif 3885   ⊆ wss 3888  {csn 4562  ^◡ccnv 5589  ran crn 5591   “ cima 5593  Fun wfun 6431   Fn wfn 6432  ⟶wf 6433  –1-1→wf1 6434  ‘cfv 6437  (class class class)co 7284   supp csupp 7986  Basecbs 16921  Scalarcsca 16974   ·_𝑠 cvsca 16975  0_gc0g 17159  Ringcrg 19792  LSpanclspn 20242  LBasisclbs 20345  NzRingcnzr 20537   freeLMod cfrlm 20962   unitVec cuvc 20998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-of 7542  df-om 7722  df-1st 7840  df-2nd 7841  df-supp 7987  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-map 8626  df-ixp 8695  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-fsupp 9138  df-sup 9210  df-oi 9278  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-nn 11983  df-2 12045  df-3 12046  df-4 12047  df-5 12048  df-6 12049  df-7 12050  df-8 12051  df-9 12052  df-n0 12243  df-z 12329  df-dec 12447  df-uz 12592  df-fz 13249  df-fzo 13392  df-seq 13731  df-hash 14054  df-struct 16857  df-sets 16874  df-slot 16892  df-ndx 16904  df-base 16922  df-ress 16951  df-plusg 16984  df-mulr 16985  df-sca 16987  df-vsca 16988  df-ip 16989  df-tset 16990  df-ple 16991  df-ds 16993  df-hom 16995  df-cco 16996  df-0g 17161  df-gsum 17162  df-prds 17167  df-pws 17169  df-mre 17304  df-mrc 17305  df-acs 17307  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-mhm 18439  df-submnd 18440  df-grp 18589  df-minusg 18590  df-sbg 18591  df-mulg 18710  df-subg 18761  df-ghm 18841  df-cntz 18932  df-cmn 19397  df-abl 19398  df-mgp 19730  df-ur 19747  df-ring 19794  df-subrg 20031  df-lmod 20134  df-lss 20203  df-lsp 20243  df-lmhm 20293  df-lbs 20346  df-sra 20443  df-rgmod 20444  df-nzr 20538  df-dsmm 20948  df-frlm 20963  df-uvc 20999

This theorem is referenced by:  frlmup3  21016  frlmup4  21017  lmisfree  21058  frlmisfrlm  21064  frlmdim  31703  lindsdom  35780  aacllem  46516