Theorem frlmlbs 21002

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 2740	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	1, 2, 3	uvcff 20996	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶(Base‘𝐹))
5	4	frnd 6606	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ⊆ (Base‘𝐹))
6		suppssdm 7984	. . . . . 6 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
7		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	2, 7, 3	frlmbasf 20965	. . . . . . 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 6618	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → (𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
11	10	ralrimiva 3110	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
12		rabid2 3313	. . . 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 3948	. . . 4 ⊢ 𝐼 ⊆ 𝐼
15		eqid 2740	. . . . 5 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
16		eqid 2740	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
17		eqid 2740	. . . . 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 21001	. . . 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 6598	. . . . 5 ⊢ (𝑈:𝐼⟶(Base‘𝐹) → 𝑈 Fn 𝐼)
21		fnima 6561	. . . . 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 6775	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = ((LSpan‘𝐹)‘ran 𝑈))
24	13, 19, 23	3eqtr2rd 2787	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹))
25		eqid 2740	. . . . . 6 ⊢ ( ·𝑠 ‘𝐹) = ( ·𝑠 ‘𝐹)
26		eqid 2740	. . . . . 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 4072	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ {𝑐}) ⊆ 𝐼)
30		vsnid 4604	. . . . . . 7 ⊢ 𝑐 ∈ {𝑐}
31		snssi 4747	. . . . . . . . 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 4198	. . . . . . . 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 2848	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ (𝐼 ∖ (𝐼 ∖ {𝑐})))
36	2	frlmsca 20958	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
37	36	fveq2d 6775	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
38	36	fveq2d 6775	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g‘𝑅) = (0g‘(Scalar‘𝐹)))
39	38	sneqd 4579	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → {(0g‘𝑅)} = {(0g‘(Scalar‘𝐹))})
40	37, 39	difeq12d 4063	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))
41	40	eleq2d 2826	. . . . . . . 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 21000	. . . . 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 20535	. . . . . . . . . . 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 20997	. . . . . . . . . 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 6437	. . . . . . . . . 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 6516	. . . . . . . . 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 6669	. . . . . . . . . 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 6844	. . . . . . . . . . 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 2746	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ {𝑐}) = {(𝑈‘𝑐)})
60	54, 59	difeq12d 4063	. . . . . . . 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 2781	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (ran 𝑈 ∖ {(𝑈‘𝑐)}) = (𝑈 “ (𝐼 ∖ {𝑐})))
62	61	fveq2d 6775	. . . . . 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 21001	. . . . . . 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 2780	. . . . 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 2863	. . . 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 3117	. . 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 7279	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → (𝑏( ·𝑠 ‘𝐹)𝑎) = (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)))
69		sneq 4577	. . . . . . . . . 10 ⊢ (𝑎 = (𝑈‘𝑐) → {𝑎} = {(𝑈‘𝑐)})
70	69	difeq2d 4062	. . . . . . . . 9 ⊢ (𝑎 = (𝑈‘𝑐) → (ran 𝑈 ∖ {𝑎}) = (ran 𝑈 ∖ {(𝑈‘𝑐)}))
71	70	fveq2d 6775	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) = ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
72	68, 71	eleq12d 2835	. . . . . . 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 3123	. . . . 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 6961	. . . 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 7305	. . 3 ⊢ 𝐹 ∈ V
79		eqid 2740	. . . 4 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
80		eqid 2740	. . . 4 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
81		frlmlbs.j	. . . 4 ⊢ 𝐽 = (LBasis‘𝐹)
82		eqid 2740	. . . 4 ⊢ (0g‘(Scalar‘𝐹)) = (0g‘(Scalar‘𝐹))
83	3, 79, 25, 80, 81, 15, 82	islbs 20336	. . 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 1542   ∈ wcel 2110  ∀wral 3066  {crab 3070  Vcvv 3431   ∖ cdif 3889   ⊆ wss 3892  {csn 4567  ^◡ccnv 5589  ran crn 5591   “ cima 5593  Fun wfun 6426   Fn wfn 6427  ⟶wf 6428  –1-1→wf1 6429  ‘cfv 6432  (class class class)co 7271   supp csupp 7968  Basecbs 16910  Scalarcsca 16963   ·_𝑠 cvsca 16964  0_gc0g 17148  Ringcrg 19781  LSpanclspn 20231  LBasisclbs 20334  NzRingcnzr 20526   freeLMod cfrlm 20951   unitVec cuvc 20987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  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 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-of 7527  df-om 7707  df-1st 7824  df-2nd 7825  df-supp 7969  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-er 8481  df-map 8600  df-ixp 8669  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fsupp 9107  df-sup 9179  df-oi 9247  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12437  df-uz 12582  df-fz 13239  df-fzo 13382  df-seq 13720  df-hash 14043  df-struct 16846  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-ress 16940  df-plusg 16973  df-mulr 16974  df-sca 16976  df-vsca 16977  df-ip 16978  df-tset 16979  df-ple 16980  df-ds 16982  df-hom 16984  df-cco 16985  df-0g 17150  df-gsum 17151  df-prds 17156  df-pws 17158  df-mre 17293  df-mrc 17294  df-acs 17296  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-mhm 18428  df-submnd 18429  df-grp 18578  df-minusg 18579  df-sbg 18580  df-mulg 18699  df-subg 18750  df-ghm 18830  df-cntz 18921  df-cmn 19386  df-abl 19387  df-mgp 19719  df-ur 19736  df-ring 19783  df-subrg 20020  df-lmod 20123  df-lss 20192  df-lsp 20232  df-lmhm 20282  df-lbs 20335  df-sra 20432  df-rgmod 20433  df-nzr 20527  df-dsmm 20937  df-frlm 20952  df-uvc 20988

This theorem is referenced by:  frlmup3  21005  frlmup4  21006  lmisfree  21047  frlmisfrlm  21053  frlmdim  31690  lindsdom  35767  aacllem  46474