Theorem frlmlbs 21736

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 2733	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	1, 2, 3	uvcff 21730	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶(Base‘𝐹))
5	4	frnd 6664	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ⊆ (Base‘𝐹))
6		suppssdm 8113	. . . . . 6 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
7		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	2, 7, 3	frlmbasf 21699	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
9	8	adantll 714	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
10	6, 9	fssdm 6675	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → (𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
11	10	ralrimiva 3125	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
12		rabid2 3429	. . . 4 ⊢ ((Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼} ↔ ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g‘𝑅)) ⊆ 𝐼)
13	11, 12	sylibr 234	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
14		ssid 3953	. . . 4 ⊢ 𝐼 ⊆ 𝐼
15		eqid 2733	. . . . 5 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
16		eqid 2733	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
17		eqid 2733	. . . . 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 21735	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐼) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
19	14, 18	mp3an3 1452	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ 𝐼})
20		ffn 6656	. . . . 5 ⊢ (𝑈:𝐼⟶(Base‘𝐹) → 𝑈 Fn 𝐼)
21		fnima 6616	. . . . 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 6832	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘(𝑈 “ 𝐼)) = ((LSpan‘𝐹)‘ran 𝑈))
24	13, 19, 23	3eqtr2rd 2775	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹))
25		eqid 2733	. . . . . 6 ⊢ ( ·𝑠 ‘𝐹) = ( ·𝑠 ‘𝐹)
26		eqid 2733	. . . . . 6 ⊢ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g‘𝑅)) ⊆ (𝐼 ∖ {𝑐})}
27		simpll 766	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ Ring)
28		simplr 768	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝐼 ∈ 𝑉)
29		difssd 4086	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ {𝑐}) ⊆ 𝐼)
30		vsnid 4615	. . . . . . 7 ⊢ 𝑐 ∈ {𝑐}
31		snssi 4759	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐼 → {𝑐} ⊆ 𝐼)
32	31	ad2antrl 728	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {𝑐} ⊆ 𝐼)
33		dfss4 4218	. . . . . . . 8 ⊢ ({𝑐} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
34	32, 33	sylib 218	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
35	30, 34	eleqtrrid 2840	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ (𝐼 ∖ (𝐼 ∖ {𝑐})))
36	2	frlmsca 21692	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
37	36	fveq2d 6832	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
38	36	fveq2d 6832	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g‘𝑅) = (0g‘(Scalar‘𝐹)))
39	38	sneqd 4587	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → {(0g‘𝑅)} = {(0g‘(Scalar‘𝐹))})
40	37, 39	difeq12d 4076	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))
41	40	eleq2d 2819	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ↔ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})))
42	41	biimpar 477	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
43	42	adantrl 716	. . . . . 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 21734	. . . . 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 20440	. . . . . . . . . . 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 21731	. . . . . . . . . 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 6491	. . . . . . . . . 10 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) ↔ (𝑈:𝐼⟶(Base‘𝐹) ∧ Fun ◡𝑈))
50	49	simprbi 496	. . . . . . . . 9 ⊢ (𝑈:𝐼–1-1→(Base‘𝐹) → Fun ◡𝑈)
51		imadif 6570	. . . . . . . . 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 6725	. . . . . . . . . 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 770	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ 𝐼)
57		fnsnfv 6907	. . . . . . . . . . 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 2739	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ {𝑐}) = {(𝑈‘𝑐)})
60	54, 59	difeq12d 4076	. . . . . . . 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 2769	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (ran 𝑈 ∖ {(𝑈‘𝑐)}) = (𝑈 “ (𝐼 ∖ {𝑐})))
62	61	fveq2d 6832	. . . . . 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 21735	. . . . . . 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 1373	. . . . . 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 2768	. . . . 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 2856	. . . 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 3176	. . 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 7360	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → (𝑏( ·𝑠 ‘𝐹)𝑎) = (𝑏( ·𝑠 ‘𝐹)(𝑈‘𝑐)))
69		sneq 4585	. . . . . . . . . 10 ⊢ (𝑎 = (𝑈‘𝑐) → {𝑎} = {(𝑈‘𝑐)})
70	69	difeq2d 4075	. . . . . . . . 9 ⊢ (𝑎 = (𝑈‘𝑐) → (ran 𝑈 ∖ {𝑎}) = (ran 𝑈 ∖ {(𝑈‘𝑐)}))
71	70	fveq2d 6832	. . . . . . . 8 ⊢ (𝑎 = (𝑈‘𝑐) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) = ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈‘𝑐)})))
72	68, 71	eleq12d 2827	. . . . . . 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 3156	. . . . 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 7027	. . . 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 257	. 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 7386	. . 3 ⊢ 𝐹 ∈ V
79		eqid 2733	. . . 4 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
80		eqid 2733	. . . 4 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
81		frlmlbs.j	. . . 4 ⊢ 𝐽 = (LBasis‘𝐹)
82		eqid 2733	. . . 4 ⊢ (0g‘(Scalar‘𝐹)) = (0g‘(Scalar‘𝐹))
83	3, 79, 25, 80, 81, 15, 82	islbs 21012	. . 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 1344	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  Vcvv 3437   ∖ cdif 3895   ⊆ wss 3898  {csn 4575  ^◡ccnv 5618  ran crn 5620   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  –1-1→wf1 6483  ‘cfv 6486  (class class class)co 7352   supp csupp 8096  Basecbs 17122  Scalarcsca 17166   ·_𝑠 cvsca 17167  0_gc0g 17345  Ringcrg 20153  NzRingcnzr 20429  LSpanclspn 20906  LBasisclbs 21010   freeLMod cfrlm 21685   unitVec cuvc 21721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-sup 9333  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-hom 17187  df-cco 17188  df-0g 17347  df-gsum 17348  df-prds 17353  df-pws 17355  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-submnd 18694  df-grp 18851  df-minusg 18852  df-sbg 18853  df-mulg 18983  df-subg 19038  df-ghm 19127  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-nzr 20430  df-subrg 20487  df-lmod 20797  df-lss 20867  df-lsp 20907  df-lmhm 20958  df-lbs 21011  df-sra 21109  df-rgmod 21110  df-dsmm 21671  df-frlm 21686  df-uvc 21722

This theorem is referenced by:  frlmup3  21739  frlmup4  21740  lmisfree  21781  frlmisfrlm  21787  frlmdim  33645  lindsdom  37674  aacllem  49926