lindsenlbs - Mathbox for Brendan Leahy

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Theorem lindsenlbs 37644

Description: A maximal linearly independent set in a free module of finite dimension over a division ring is a basis. (Contributed by Brendan Leahy, 2-Jun-2021.)

**Assertion**
Ref	Expression
lindsenlbs	⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))

Proof of Theorem lindsenlbs Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl3 1194	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
2		drngring 20701	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3		eqid 2736	. . . . . . . 8 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
4	3	frlmlmod 21714	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
5	2, 4	sylan 580	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6		eqid 2736	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
7	6	linds1 21775	. . . . . 6 ⊢ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) → 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼)))
8		eqid 2736	. . . . . . 7 ⊢ (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
9	6, 8	lspssv 20945	. . . . . 6 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
10	5, 7, 9	syl2an 596	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
11	10	3impa 1109	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
12	11	adantr 480	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
13		bren2 9002	. . . . . . 7 ⊢ (𝑋 ≈ 𝐼 ↔ (𝑋 ≼ 𝐼 ∧ ¬ 𝑋 ≺ 𝐼))
14	13	simprbi 496	. . . . . 6 ⊢ (𝑋 ≈ 𝐼 → ¬ 𝑋 ≺ 𝐼)
15		snfi 9062	. . . . . . . . . . . 12 ⊢ {𝑦} ∈ Fin
16		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝐼 ∈ Fin)
17		lindsdom 37643	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ≼ 𝐼)
18		domfi 9208	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑋 ≼ 𝐼) → 𝑋 ∈ Fin)
19	16, 17, 18	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ∈ Fin)
20		unfi 9190	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ 𝑋 ∈ Fin) → ({𝑦} ∪ 𝑋) ∈ Fin)
21	15, 19, 20	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ∈ Fin)
22	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ({𝑦} ∪ 𝑋) ∈ Fin)
23		vex 3468	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
24	23	snss 4766	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑋 ↔ {𝑦} ⊆ 𝑋)
25	6, 8	lspssid 20947	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
26	5, 7, 25	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
27	26	3impa 1109	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
28	27	sseld 3962	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑦 ∈ 𝑋 → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
29	24, 28	biimtrrid 243	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ({𝑦} ⊆ 𝑋 → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
30	29	con3dimp 408	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ {𝑦} ⊆ 𝑋)
31		nsspssun 4248	. . . . . . . . . . 11 ⊢ (¬ {𝑦} ⊆ 𝑋 ↔ 𝑋 ⊊ ({𝑦} ∪ 𝑋))
32	30, 31	sylib 218	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → 𝑋 ⊊ ({𝑦} ∪ 𝑋))
33		php3 9228	. . . . . . . . . 10 ⊢ ((({𝑦} ∪ 𝑋) ∈ Fin ∧ 𝑋 ⊊ ({𝑦} ∪ 𝑋)) → 𝑋 ≺ ({𝑦} ∪ 𝑋))
34	22, 32, 33	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → 𝑋 ≺ ({𝑦} ∪ 𝑋))
35	34	adantrl 716	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → 𝑋 ≺ ({𝑦} ∪ 𝑋))
36		simpl1 1192	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → 𝑅 ∈ DivRing)
37		simpl2 1193	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → 𝐼 ∈ Fin)
38		snssi 4789	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) → {𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)))
39	38	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → {𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)))
40	7	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼)))
41		unss 4170	. . . . . . . . . . . 12 ⊢ (({𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) ↔ ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
42	41	biimpi 216	. . . . . . . . . . 11 ⊢ (({𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
43	39, 40, 42	syl2anr 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
44		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
45	28	con3dimp 408	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ 𝑦 ∈ 𝑋)
46		difsn 4779	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 ∈ 𝑋 → (𝑋 ∖ {𝑦}) = 𝑋)
47	45, 46	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → (𝑋 ∖ {𝑦}) = 𝑋)
48	47	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})) = ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
49	44, 48	neleqtrrd 2858	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})))
50	49	adantlr 715	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})))
51		difsnid 4791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑋 → ((𝑋 ∖ {𝑧}) ∪ {𝑧}) = 𝑋)
52	51	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑋 → ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})) = ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
53	52	eleq2d 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑋 → (𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})) ↔ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
54	53	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑋 → (¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})) ↔ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
55	54	biimparc 479	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})))
56	55	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})))
57	3	frlmsca 21718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
58		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ DivRing)
59	57, 58	eqeltrrd 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ DivRing)
60		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
61	60	islvec 21067	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 freeLMod 𝐼) ∈ LVec ↔ ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ DivRing))
62	5, 59, 61	sylanbrc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LVec)
63	62	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑅 freeLMod 𝐼) ∈ LVec)
64	63	ad4antr 732	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → (𝑅 freeLMod 𝐼) ∈ LVec)
65	7	ssdifssd 4127	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) → (𝑋 ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
66	65	3ad2ant3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑋 ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
67	66	ad4antr 732	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → (𝑋 ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
68		simp-4r 783	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)))
69		difundir 4271	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑧}) = (({𝑦} ∖ {𝑧}) ∪ (𝑋 ∖ {𝑧}))
70	69	equncomi 4140	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑧}) = ((𝑋 ∖ {𝑧}) ∪ ({𝑦} ∖ {𝑧}))
71		elsni 4623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∈ {𝑦} → 𝑧 = 𝑦)
72	71	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ {𝑦} → (𝑧 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
73	72	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ {𝑦} → (¬ 𝑧 ∈ 𝑋 ↔ ¬ 𝑦 ∈ 𝑋))
74	45, 73	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → (𝑧 ∈ {𝑦} → ¬ 𝑧 ∈ 𝑋))
75	74	con2d 134	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → (𝑧 ∈ 𝑋 → ¬ 𝑧 ∈ {𝑦}))
76	75	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ {𝑦})
77		difsn 4779	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑧 ∈ {𝑦} → ({𝑦} ∖ {𝑧}) = {𝑦})
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ({𝑦} ∖ {𝑧}) = {𝑦})
79	78	uneq2d 4148	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ((𝑋 ∖ {𝑧}) ∪ ({𝑦} ∖ {𝑧})) = ((𝑋 ∖ {𝑧}) ∪ {𝑦}))
80	70, 79	eqtrid 2783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → (({𝑦} ∪ 𝑋) ∖ {𝑧}) = ((𝑋 ∖ {𝑧}) ∪ {𝑦}))
81	80	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) = ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑦})))
82	81	eleq2d 2821	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑦}))))
83	82	adantllr 719	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑦}))))
84	83	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑦})))
85		drngnzr 20713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
86	85	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
87	57, 86	eqeltrrd 2836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
88	5, 87	jca 511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
89	88	anim1i 615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))))
90	89	3impa 1109	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))))
91	8, 60	lindsind2 21784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧})))
92	91	3expa 1118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧})))
93	90, 92	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧})))
94	93	ad5ant14 757	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧})))
95	84, 94	eldifd 3942	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → 𝑧 ∈ (((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑦})) ∖ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧}))))
96		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (LSubSp‘(𝑅 freeLMod 𝐼)) = (LSubSp‘(𝑅 freeLMod 𝐼))
97	6, 96, 8	lspsolv 21109	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 freeLMod 𝐼) ∈ LVec ∧ ((𝑋 ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ 𝑧 ∈ (((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑦})) ∖ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧}))))) → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})))
98	64, 67, 68, 95, 97	syl13anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})))
99	56, 98	mtand 815	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})))
100	99	ralrimiva 3133	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ∀𝑧 ∈ 𝑋 ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})))
101		ralunb 4177	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ ({𝑦} ∪ 𝑋) ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ (∀𝑧 ∈ {𝑦} ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∧ ∀𝑧 ∈ 𝑋 ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
102		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
103		sneq 4616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
104	103	difeq2d 4106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (({𝑦} ∪ 𝑋) ∖ {𝑧}) = (({𝑦} ∪ 𝑋) ∖ {𝑦}))
105		uncom 4138	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({𝑦} ∪ 𝑋) = (𝑋 ∪ {𝑦})
106	105	difeq1i 4102	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑦}) = ((𝑋 ∪ {𝑦}) ∖ {𝑦})
107		difun2 4461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∪ {𝑦}) ∖ {𝑦}) = (𝑋 ∖ {𝑦})
108	106, 107	eqtri 2759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑦}) = (𝑋 ∖ {𝑦})
109	104, 108	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → (({𝑦} ∪ 𝑋) ∖ {𝑧}) = (𝑋 ∖ {𝑦}))
110	109	fveq2d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) = ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})))
111	102, 110	eleq12d 2829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦}))))
112	111	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦}))))
113	23, 112	ralsn 4662	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ {𝑦} ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})))
114	113	anbi1i 624	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑧 ∈ {𝑦} ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∧ ∀𝑧 ∈ 𝑋 ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))) ↔ (¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})) ∧ ∀𝑧 ∈ 𝑋 ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
115	101, 114	bitri 275	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ({𝑦} ∪ 𝑋) ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ (¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})) ∧ ∀𝑧 ∈ 𝑋 ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
116	50, 100, 115	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ∀𝑧 ∈ ({𝑦} ∪ 𝑋) ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})))
117	116	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → (¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) → ∀𝑧 ∈ ({𝑦} ∪ 𝑋) ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
118	63	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (𝑅 freeLMod 𝐼) ∈ LVec)
119		eldifsn 4767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ↔ (𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∧ 𝑥 ≠ (0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))))
120	119	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) → (𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∧ 𝑥 ≠ (0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))))
121	120	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∧ 𝑥 ≠ (0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))))
122	38, 7, 42	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
123	122	3ad2antl3 1188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
124	123	sselda 3963	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → 𝑧 ∈ (Base‘(𝑅 freeLMod 𝐼)))
125	124	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → 𝑧 ∈ (Base‘(𝑅 freeLMod 𝐼)))
126		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·_𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·_𝑠 ‘(𝑅 freeLMod 𝐼))
127		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
128		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0_g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))
129	6, 60, 126, 127, 128, 8	lspsnvs 21080	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 freeLMod 𝐼) ∈ LVec ∧ (𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∧ 𝑥 ≠ (0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))) ∧ 𝑧 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘{(𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧)}) = ((LSpan‘(𝑅 freeLMod 𝐼))‘{𝑧}))
130	118, 121, 125, 129	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → ((LSpan‘(𝑅 freeLMod 𝐼))‘{(𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧)}) = ((LSpan‘(𝑅 freeLMod 𝐼))‘{𝑧}))
131	130	sseq1d 3995	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (((LSpan‘(𝑅 freeLMod 𝐼))‘{(𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧)}) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ((LSpan‘(𝑅 freeLMod 𝐼))‘{𝑧}) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
132	5	3adant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑅 freeLMod 𝐼) ∈ LMod)
133	132	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (𝑅 freeLMod 𝐼) ∈ LMod)
134		df-3an 1088	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ↔ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))))
135	122	ssdifssd 4127	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → (({𝑦} ∪ 𝑋) ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
136	6, 96, 8	lspcl 20938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (({𝑦} ∪ 𝑋) ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
137	5, 135, 136	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
138	137	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
139	134, 138	sylanb 581	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
140	139	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
141		eldifi 4111	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) → 𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
142	141	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → 𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
143	6, 60, 126, 127	lmodvscl 20840	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ (Base‘(𝑅 freeLMod 𝐼))) → (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ (Base‘(𝑅 freeLMod 𝐼)))
144	133, 142, 125, 143	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ (Base‘(𝑅 freeLMod 𝐼)))
145	6, 96, 8, 133, 140, 144	ellspsn5b 20957	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → ((𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ((LSpan‘(𝑅 freeLMod 𝐼))‘{(𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧)}) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
146	132	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → (𝑅 freeLMod 𝐼) ∈ LMod)
147	139	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
148	6, 96, 8, 146, 147, 124	ellspsn5b 20957	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ((LSpan‘(𝑅 freeLMod 𝐼))‘{𝑧}) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
149	148	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ((LSpan‘(𝑅 freeLMod 𝐼))‘{𝑧}) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
150	131, 145, 149	3bitr4rd 312	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
151	150	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
152	151	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) ∧ 𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))})) → (¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) → ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
153	152	ralrimdva 3141	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → (¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) → ∀𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
154	153	ralimdva 3153	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → (∀𝑧 ∈ ({𝑦} ∪ 𝑋) ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) → ∀𝑧 ∈ ({𝑦} ∪ 𝑋)∀𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
155	117, 154	syld 47	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → (¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) → ∀𝑧 ∈ ({𝑦} ∪ 𝑋)∀𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
156	155	impr 454	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → ∀𝑧 ∈ ({𝑦} ∪ 𝑋)∀𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})))
157		ovex 7443	. . . . . . . . . . 11 ⊢ (𝑅 freeLMod 𝐼) ∈ V
158	6, 126, 8, 60, 127, 128	islinds2 21778	. . . . . . . . . . 11 ⊢ ((𝑅 freeLMod 𝐼) ∈ V → (({𝑦} ∪ 𝑋) ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ↔ (({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ ∀𝑧 ∈ ({𝑦} ∪ 𝑋)∀𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})))))
159	157, 158	ax-mp 5	. . . . . . . . . 10 ⊢ (({𝑦} ∪ 𝑋) ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ↔ (({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ ∀𝑧 ∈ ({𝑦} ∪ 𝑋)∀𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) ¬ (𝑥( ·_𝑠 ‘(𝑅 freeLMod 𝐼))𝑧) ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧}))))
160	43, 156, 159	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → ({𝑦} ∪ 𝑋) ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
161		lindsdom 37643	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ ({𝑦} ∪ 𝑋) ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ≼ 𝐼)
162	36, 37, 160, 161	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → ({𝑦} ∪ 𝑋) ≼ 𝐼)
163		sdomdomtr 9129	. . . . . . . 8 ⊢ ((𝑋 ≺ ({𝑦} ∪ 𝑋) ∧ ({𝑦} ∪ 𝑋) ≼ 𝐼) → 𝑋 ≺ 𝐼)
164	35, 162, 163	syl2anc 584	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → 𝑋 ≺ 𝐼)
165	164	stoic1a 1772	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑋 ≺ 𝐼) → ¬ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
166	14, 165	sylan2 593	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → ¬ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
167		iman 401	. . . . 5 ⊢ ((𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ↔ ¬ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
168	166, 167	sylibr 234	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
169	168	ssrdv 3969	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → (Base‘(𝑅 freeLMod 𝐼)) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
170	12, 169	eqssd 3981	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) = (Base‘(𝑅 freeLMod 𝐼)))
171		eqid 2736	. . 3 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
172	6, 171, 8	islbs4 21797	. 2 ⊢ (𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)) ↔ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) = (Base‘(𝑅 freeLMod 𝐼))))
173	1, 170, 172	sylanbrc 583	1 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 Vcvv 3464 ∖ cdif 3928 ∪ cun 3929 ⊆ wss 3931 ⊊ wpss 3932 {csn 4606 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ≈ cen 8961 ≼ cdom 8962 ≺ csdm 8963 Fincfn 8964 Basecbs 17233 Scalarcsca 17279 ·_𝑠 cvsca 17280 0_gc0g 17458 Ringcrg 20198 NzRingcnzr 20477 DivRingcdr 20694 LModclmod 20822 LSubSpclss 20893 LSpanclspn 20933 LBasisclbs 21037 LVecclvec 21065 freeLMod cfrlm 21711 LIndSclinds 21770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17603 df-mrc 17604 df-mri 17605 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-nzr 20478 df-subrg 20535 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-lmhm 20985 df-lbs 21038 df-lvec 21066 df-sra 21136 df-rgmod 21137 df-dsmm 21697 df-frlm 21712 df-uvc 21748 df-lindf 21771 df-linds 21772

This theorem is referenced by: matunitlindflem2 37646

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