lindsenlbs - Mathbox for Brendan Leahy

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Theorem lindsenlbs 38153

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 1210	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)))
2		drngring 20819	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3		eqid 2769	. . . . . . . 8 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
4	3	frlmlmod 21867	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
5	2, 4	sylan 591	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LMod)
6		eqid 2769	. . . . . . 7 ⊢ (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
7	6	linds1 21928	. . . . . 6 ⊢ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) → 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼)))
8		eqid 2769	. . . . . . 7 ⊢ (LSpan‘(𝑅 freeLMod 𝐼)) = (LSpan‘(𝑅 freeLMod 𝐼))
9	6, 8	lspssv 21081	. . . . . 6 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
10	5, 7, 9	syl2an 607	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
11	10	3impa 1125	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
12	11	adantr 485	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
13		bren2 8979	. . . . . . 7 ⊢ (𝑋 ≈ 𝐼 ↔ (𝑋 ≼ 𝐼 ∧ ¬ 𝑋 ≺ 𝐼))
14	13	simprbi 502	. . . . . 6 ⊢ (𝑋 ≈ 𝐼 → ¬ 𝑋 ≺ 𝐼)
15		snfi 9039	. . . . . . . . . . . 12 ⊢ {𝑦} ∈ Fin
16		simp2 1153	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝐼 ∈ Fin)
17		lindsdom 38152	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ≼ 𝐼)
18		domfi 9172	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑋 ≼ 𝐼) → 𝑋 ∈ Fin)
19	16, 17, 18	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ∈ Fin)
20		unfi 9154	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ 𝑋 ∈ Fin) → ({𝑦} ∪ 𝑋) ∈ Fin)
21	15, 19, 20	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ∈ Fin)
22	21	adantr 485	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ({𝑦} ∪ 𝑋) ∈ Fin)
23		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
24	23	snss 4755	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑋 ↔ {𝑦} ⊆ 𝑋)
25	6, 8	lspssid 21083	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
26	5, 7, 25	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
27	26	3impa 1125	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
28	27	sseld 3944	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑦 ∈ 𝑋 → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
29	24, 28	biimtrrid 246	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ({𝑦} ⊆ 𝑋 → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
30	29	con3dimp 413	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ {𝑦} ⊆ 𝑋)
31		nsspssun 4229	. . . . . . . . . . 11 ⊢ (¬ {𝑦} ⊆ 𝑋 ↔ 𝑋 ⊊ ({𝑦} ∪ 𝑋))
32	30, 31	sylib 221	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → 𝑋 ⊊ ({𝑦} ∪ 𝑋))
33		php3 9192	. . . . . . . . . 10 ⊢ ((({𝑦} ∪ 𝑋) ∈ Fin ∧ 𝑋 ⊊ ({𝑦} ∪ 𝑋)) → 𝑋 ≺ ({𝑦} ∪ 𝑋))
34	22, 32, 33	syl2anc 595	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → 𝑋 ≺ ({𝑦} ∪ 𝑋))
35	34	adantrl 728	. . . . . . . 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 1208	. . . . . . . . 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 1209	. . . . . . . . 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 4756	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) → {𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)))
39	38	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → {𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)))
40	7	3ad2ant3 1151	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼)))
41		unss 4151	. . . . . . . . . . . 12 ⊢ (({𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) ↔ ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
42	41	biimpi 219	. . . . . . . . . . 11 ⊢ (({𝑦} ⊆ (Base‘(𝑅 freeLMod 𝐼)) ∧ 𝑋 ⊆ (Base‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
43	39, 40, 42	syl2anr 608	. . . . . . . . . 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 489	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → ¬ 𝑦 ∈ 𝑋)
46		difsn 4770	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 ∈ 𝑋 → (𝑋 ∖ {𝑦}) = 𝑋)
47	45, 46	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → (𝑋 ∖ {𝑦}) = 𝑋)
48	47	fveq2d 6886	. . . . . . . . . . . . . . . 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 2892	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 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 4780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑋 → ((𝑋 ∖ {𝑧}) ∪ {𝑧}) = 𝑋)
52	51	fveq2d 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑋 → ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})) = ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
53	52	eleq2d 2855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑋 → (𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})) ↔ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
54	53	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑋 → (¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})) ↔ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
55	54	biimparc 484	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘((𝑋 ∖ {𝑧}) ∪ {𝑧})))
56	55	adantll 726	. . . . . . . . . . . . . . . 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 21871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod 𝐼)))
58		simpl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ DivRing)
59	57, 58	eqeltrrd 2870	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ DivRing)
60		eqid 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Scalar‘(𝑅 freeLMod 𝐼)) = (Scalar‘(𝑅 freeLMod 𝐼))
61	60	islvec 21202	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 freeLMod 𝐼) ∈ LVec ↔ ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ DivRing))
62	5, 59, 61	sylanbrc 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (𝑅 freeLMod 𝐼) ∈ LVec)
63	62	3adant3 1148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑅 freeLMod 𝐼) ∈ LVec)
64	63	ad4antr 744	. . . . . . . . . . . . . . . . 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 4109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) → (𝑋 ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
66	65	3ad2ant3 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑋 ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
67	66	ad4antr 744	. . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . 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 4252	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑧}) = (({𝑦} ∖ {𝑧}) ∪ (𝑋 ∖ {𝑧}))
70	69	equncomi 4122	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑧}) = ((𝑋 ∖ {𝑧}) ∪ ({𝑦} ∖ {𝑧}))
71		elsni 4611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∈ {𝑦} → 𝑧 = 𝑦)
72	71	eleq1d 2854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ {𝑦} → (𝑧 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
73	72	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ {𝑦} → (¬ 𝑧 ∈ 𝑋 ↔ ¬ 𝑦 ∈ 𝑋))
74	45, 73	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → (𝑧 ∈ {𝑦} → ¬ 𝑧 ∈ 𝑋))
75	74	con2d 135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) → (𝑧 ∈ 𝑋 → ¬ 𝑧 ∈ {𝑦}))
76	75	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ {𝑦})
77		difsn 4770	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑧 ∈ {𝑦} → ({𝑦} ∖ {𝑧}) = {𝑦})
78	76, 77	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ∧ 𝑧 ∈ 𝑋) → ({𝑦} ∖ {𝑧}) = {𝑦})
79	78	uneq2d 4130	. . . . . . . . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 20831	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
86	85	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → 𝑅 ∈ NzRing)
87	57, 86	eqeltrrd 2870	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing)
88	5, 87	jca 520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) → ((𝑅 freeLMod 𝐼) ∈ LMod ∧ (Scalar‘(𝑅 freeLMod 𝐼)) ∈ NzRing))
89	88	anim1i 626	. . . . . . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . . . . . 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 21937	. . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ 𝑋) → ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑧})))
94	93	ad5ant14 769	. . . . . . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (LSubSp‘(𝑅 freeLMod 𝐼)) = (LSubSp‘(𝑅 freeLMod 𝐼))
97	6, 96, 8	lspsolv 21244	. . . . . . . . . . . . . . . . 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 1397	. . . . . . . . . . . . . . . 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 827	. . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . 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 4158	. . . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
103		sneq 4604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
104	103	difeq2d 4089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (({𝑦} ∪ 𝑋) ∖ {𝑧}) = (({𝑦} ∪ 𝑋) ∖ {𝑦}))
105		uncom 4120	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({𝑦} ∪ 𝑋) = (𝑋 ∪ {𝑦})
106	105	difeq1i 4085	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑦}) = ((𝑋 ∪ {𝑦}) ∖ {𝑦})
107		difun2 4447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∪ {𝑦}) ∖ {𝑦}) = (𝑋 ∖ {𝑦})
108	106, 107	eqtri 2792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (({𝑦} ∪ 𝑋) ∖ {𝑦}) = (𝑋 ∖ {𝑦})
109	104, 108	eqtrdi 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑦 → (({𝑦} ∪ 𝑋) ∖ {𝑧}) = (𝑋 ∖ {𝑦}))
110	109	fveq2d 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) = ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})))
111	102, 110	eleq12d 2863	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦}))))
112	111	notbid 321	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦}))))
113	23, 112	ralsn 4652	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ {𝑦} ¬ 𝑧 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ↔ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘(𝑋 ∖ {𝑦})))
114	113	anbi1i 635	. . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . . 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 4758	. . . . . . . . . . . . . . . . . . . 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)))))
120	119	bilani 509	. . . . . . . . . . . . . . . . . . 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)))))
121	38, 7, 42	syl2anr 608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
122	121	3ad2antl3 1204	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
123	122	sselda 3945	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → 𝑧 ∈ (Base‘(𝑅 freeLMod 𝐼)))
124	123	adantr 485	. . . . . . . . . . . . . . . . . . 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)))
125		eqid 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·_𝑠 ‘(𝑅 freeLMod 𝐼)) = ( ·_𝑠 ‘(𝑅 freeLMod 𝐼))
126		eqid 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))) = (Base‘(Scalar‘(𝑅 freeLMod 𝐼)))
127		eqid 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0_g‘(Scalar‘(𝑅 freeLMod 𝐼))) = (0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))
128	6, 60, 125, 126, 127, 8	lspsnvs 21215	. . . . . . . . . . . . . . . . . . 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}))
129	118, 120, 124, 128	syl3anc 1396	. . . . . . . . . . . . . . . . . 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}))
130	129	sseq1d 3976	. . . . . . . . . . . . . . . . 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}))))
131	5	3adant3 1148	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → (𝑅 freeLMod 𝐼) ∈ LMod)
132	131	ad3antrrr 742	. . . . . . . . . . . . . . . . . 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)
133		df-3an 1103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ↔ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin) ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))))
134	121	ssdifssd 4109	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) → (({𝑦} ∪ 𝑋) ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼)))
135	6, 96, 8	lspcl 21074	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 freeLMod 𝐼) ∈ LMod ∧ (({𝑦} ∪ 𝑋) ∖ {𝑧}) ⊆ (Base‘(𝑅 freeLMod 𝐼))) → ((LSpan‘(𝑅 freeLMod 𝐼))‘(({𝑦} ∪ 𝑋) ∖ {𝑧})) ∈ (LSubSp‘(𝑅 freeLMod 𝐼)))
136	5, 134, 135	syl2an 607	. . . . . . . . . . . . . . . . . . . . 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)))
137	136	anassrs 472	. . . . . . . . . . . . . . . . . . . 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)))
138	133, 137	sylanb 592	. . . . . . . . . . . . . . . . . . 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)))
139	138	ad2antrr 738	. . . . . . . . . . . . . . . . . 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)))
140		eldifi 4093	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((Base‘(Scalar‘(𝑅 freeLMod 𝐼))) ∖ {(0_g‘(Scalar‘(𝑅 freeLMod 𝐼)))}) → 𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod 𝐼))))
141	140	adantl 486	. . . . . . . . . . . . . . . . . . 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))))
142	6, 60, 125, 126	lmodvscl 20976	. . . . . . . . . . . . . . . . . . 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)))
143	132, 141, 124, 142	syl3anc 1396	. . . . . . . . . . . . . . . . . 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)))
144	6, 96, 8, 132, 139, 143	ellspsn5b 21093	. . . . . . . . . . . . . . . . 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}))))
145	131	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼))) ∧ 𝑧 ∈ ({𝑦} ∪ 𝑋)) → (𝑅 freeLMod 𝐼) ∈ LMod)
146	138	adantr 485	. . . . . . . . . . . . . . . . . . 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)))
147	6, 96, 8, 145, 146, 123	ellspsn5b 21093	. . . . . . . . . . . . . . . . . 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}))))
148	147	adantr 485	. . . . . . . . . . . . . . . . 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}))))
149	130, 144, 148	3bitr4rd 315	. . . . . . . . . . . . . . . 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}))))
150	149	notbid 321	. . . . . . . . . . . . . . 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}))))
151	150	biimpd 232	. . . . . . . . . . . . . 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}))))
152	151	ralrimdva 3171	. . . . . . . . . . . . 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}))))
153	152	ralimdva 3183	. . . . . . . . . . . 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}))))
154	117, 153	syld 48	. . . . . . . . . . 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}))))
155	154	impr 459	. . . . . . . . . 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})))
156		ovex 7444	. . . . . . . . . . 11 ⊢ (𝑅 freeLMod 𝐼) ∈ V
157	6, 125, 8, 60, 126, 127	islinds2 21931	. . . . . . . . . . 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})))))
158	156, 157	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}))))
159	43, 155, 158	sylanbrc 594	. . . . . . . . 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)))
160		lindsdom 38152	. . . . . . . . 9 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ ({𝑦} ∪ 𝑋) ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → ({𝑦} ∪ 𝑋) ≼ 𝐼)
161	36, 37, 159, 160	syl3anc 1396	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → ({𝑦} ∪ 𝑋) ≼ 𝐼)
162		sdomdomtr 9097	. . . . . . . 8 ⊢ ((𝑋 ≺ ({𝑦} ∪ 𝑋) ∧ ({𝑦} ∪ 𝑋) ≼ 𝐼) → 𝑋 ≺ 𝐼)
163	35, 161, 162	syl2anc 595	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))) → 𝑋 ≺ 𝐼)
164	163	stoic1a 1799	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ ¬ 𝑋 ≺ 𝐼) → ¬ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
165	14, 164	sylan2 604	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → ¬ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
166		iman 406	. . . . 5 ⊢ ((𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)) ↔ ¬ (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) ∧ ¬ 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
167	165, 166	sylibr 237	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → (𝑦 ∈ (Base‘(𝑅 freeLMod 𝐼)) → 𝑦 ∈ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋)))
168	167	ssrdv 3951	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → (Base‘(𝑅 freeLMod 𝐼)) ⊆ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋))
169	12, 168	eqssd 3962	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) = (Base‘(𝑅 freeLMod 𝐼)))
170		eqid 2769	. . 3 ⊢ (LBasis‘(𝑅 freeLMod 𝐼)) = (LBasis‘(𝑅 freeLMod 𝐼))
171	6, 170, 8	islbs4 21950	. 2 ⊢ (𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)) ↔ (𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼)) ∧ ((LSpan‘(𝑅 freeLMod 𝐼))‘𝑋) = (Base‘(𝑅 freeLMod 𝐼))))
172	1, 169, 171	sylanbrc 594	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 Vcvv 3463 ∖ cdif 3910 ∪ cun 3911 ⊆ wss 3913 ⊊ wpss 3914 {csn 4594 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ≈ cen 8939 ≼ cdom 8940 ≺ csdm 8941 Fincfn 8942 Basecbs 17268 Scalarcsca 17312 ·_𝑠 cvsca 17313 0_gc0g 17491 Ringcrg 20314 NzRingcnzr 20594 DivRingcdr 20812 LModclmod 20958 LSubSpclss 21029 LSpanclspn 21069 LBasisclbs 21172 LVecclvec 21200 freeLMod cfrlm 21864 LIndSclinds 21923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-fzo 13682 df-seq 14037 df-hash 14366 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-hom 17333 df-cco 17334 df-0g 17493 df-gsum 17494 df-prds 17499 df-pws 17501 df-mre 17637 df-mrc 17638 df-mri 17639 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-ghm 19283 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-nzr 20595 df-subrg 20654 df-drng 20814 df-lmod 20960 df-lss 21030 df-lsp 21070 df-lmhm 21120 df-lbs 21173 df-lvec 21201 df-sra 21271 df-rgmod 21272 df-dsmm 21850 df-frlm 21865 df-uvc 21901 df-lindf 21924 df-linds 21925

This theorem is referenced by: matunitlindflem2 38155

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