lindsadd - Mathbox for Brendan Leahy

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Theorem lindsadd 38741

Description: In a vector space, the union of an independent set and a vector not in its span is an independent set. (Contributed by Brendan Leahy, 4-Mar-2023.)

**Assertion**
Ref	Expression
lindsadd	⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∪ {𝑋}) ∈ (LIndS‘𝑊))

Proof of Theorem lindsadd Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	linds1 22149	. . . 4 ⊢ (𝐹 ∈ (LIndS‘𝑊) → 𝐹 ⊆ (Base‘𝑊))
3		eldifi 4070	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹)) → 𝑋 ∈ (Base‘𝑊))
4	3	snssd 4747	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹)) → {𝑋} ⊆ (Base‘𝑊))
5		unss 4191	. . . . 5 ⊢ ((𝐹 ⊆ (Base‘𝑊) ∧ {𝑋} ⊆ (Base‘𝑊)) ↔ (𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊))
6	5	biimpi 219	. . . 4 ⊢ ((𝐹 ⊆ (Base‘𝑊) ∧ {𝑋} ⊆ (Base‘𝑊)) → (𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊))
7	2, 4, 6	syl2an 608	. . 3 ⊢ ((𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊))
8	7	3adant1 1148	. 2 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊))
9		eldifn 4071	. . . . . . 7 ⊢ (𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹)) → ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹))
10	9	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹))
11	10	adantr 486	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹))
12		simpll1 1231	. . . . . . . 8 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → 𝑊 ∈ LVec)
13	2	ssdifssd 4086	. . . . . . . . . 10 ⊢ (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ∖ {𝑥}) ⊆ (Base‘𝑊))
14	13	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∖ {𝑥}) ⊆ (Base‘𝑊))
15	14	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → (𝐹 ∖ {𝑥}) ⊆ (Base‘𝑊))
16	3	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → 𝑋 ∈ (Base‘𝑊))
17	16	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → 𝑋 ∈ (Base‘𝑊))
18		simpr 490	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})))
19		lveclmod 21414	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
20	19	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑊 ∈ LMod)
21		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
22	21	lmodring 21176	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
23	19, 22	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ Ring)
24		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
25		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26	24, 25	ringelnzr 20807	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑊) ∈ Ring ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → (Scalar‘𝑊) ∈ NzRing)
27	23, 26	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LVec ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → (Scalar‘𝑊) ∈ NzRing)
28	27	ad2ant2rl 762	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (Scalar‘𝑊) ∈ NzRing)
29		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝐹 ∈ (LIndS‘𝑊))
30		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑥 ∈ 𝐹)
31		eqid 2763	. . . . . . . . . . . . 13 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
32	31, 21	lindsind2 22158	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑥 ∈ 𝐹) → ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑥})))
33	20, 28, 29, 30, 32	syl211anc 1404	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑥})))
34	33	3adantl3 1187	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑥})))
35	34	adantr 486	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑥})))
36	18, 35	eldifd 4053	. . . . . . . 8 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → 𝑥 ∈ (((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) ∖ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑥}))))
37		eqid 2763	. . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
38	1, 37, 31	lspsolv 21454	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ ((𝐹 ∖ {𝑥}) ⊆ (Base‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) ∖ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑥}))))) → 𝑋 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑥})))
39	12, 15, 17, 36, 38	syl13anc 1400	. . . . . . 7 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) ∧ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋}))) → 𝑋 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑥})))
40	39	ex 418	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) → 𝑋 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑥}))))
41		eldif 4052	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹)) ↔ (𝑋 ∈ (Base‘𝑊) ∧ ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
42		snssi 4746	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∈ 𝐹 → {𝑋} ⊆ 𝐹)
43	1, 31	lspss 21292	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 ⊆ (Base‘𝑊) ∧ {𝑋} ⊆ 𝐹) → ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹))
44	19, 2, 42, 43	syl3an 1178	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ 𝐹) → ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹))
45	44	ad4ant124 1192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑋 ∈ 𝐹) → ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹))
46	1, 31	lspsnid 21301	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋 ∈ ((LSpan‘𝑊)‘{𝑋}))
47	19, 46	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋 ∈ ((LSpan‘𝑊)‘{𝑋}))
48	47	ad4ant13 764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑋 ∈ 𝐹) → 𝑋 ∈ ((LSpan‘𝑊)‘{𝑋}))
49	45, 48	sseldd 3914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑋 ∈ 𝐹) → 𝑋 ∈ ((LSpan‘𝑊)‘𝐹))
50	49	ex 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑋 ∈ 𝐹 → 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
51	50	con3d 153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ 𝑋 ∈ (Base‘𝑊)) → (¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹) → ¬ 𝑋 ∈ 𝐹))
52	51	expimpd 459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) → ((𝑋 ∈ (Base‘𝑊) ∧ ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)) → ¬ 𝑋 ∈ 𝐹))
53	52	3impia 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ (𝑋 ∈ (Base‘𝑊) ∧ ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹))) → ¬ 𝑋 ∈ 𝐹)
54	41, 53	syl3an3b 1433	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ¬ 𝑋 ∈ 𝐹)
55		eleq1 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = 𝑥 → (𝑋 ∈ 𝐹 ↔ 𝑥 ∈ 𝐹))
56	55	notbid 321	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = 𝑥 → (¬ 𝑋 ∈ 𝐹 ↔ ¬ 𝑥 ∈ 𝐹))
57	54, 56	syl5ibcom 248	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝑋 = 𝑥 → ¬ 𝑥 ∈ 𝐹))
58	57	necon2ad 2974	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝑥 ∈ 𝐹 → 𝑋 ≠ 𝑥))
59	58	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → 𝑋 ≠ 𝑥)
60		disjsn2 4673	. . . . . . . . . . . . . 14 ⊢ (𝑋 ≠ 𝑥 → ({𝑋} ∩ {𝑥}) = ∅)
61	59, 60	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → ({𝑋} ∩ {𝑥}) = ∅)
62		disj3 4410	. . . . . . . . . . . . 13 ⊢ (({𝑋} ∩ {𝑥}) = ∅ ↔ {𝑋} = ({𝑋} ∖ {𝑥}))
63	61, 62	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → {𝑋} = ({𝑋} ∖ {𝑥}))
64	63	uneq2d 4170	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → ((𝐹 ∖ {𝑥}) ∪ {𝑋}) = ((𝐹 ∖ {𝑥}) ∪ ({𝑋} ∖ {𝑥})))
65		difundir 4239	. . . . . . . . . . 11 ⊢ ((𝐹 ∪ {𝑋}) ∖ {𝑥}) = ((𝐹 ∖ {𝑥}) ∪ ({𝑋} ∖ {𝑥}))
66	64, 65	eqtr4di 2816	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → ((𝐹 ∖ {𝑥}) ∪ {𝑋}) = ((𝐹 ∪ {𝑋}) ∖ {𝑥}))
67	66	fveq2d 6915	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) = ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))
68	67	eleq2d 2850	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) ↔ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
69	68	adantrr 730	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) ↔ 𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
70		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LVec ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → 𝑊 ∈ LVec)
71		eldifsn 4748	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ↔ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))))
72	71	bilani 510	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LVec ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))))
73	2	sselda 3913	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (LIndS‘𝑊) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (Base‘𝑊))
74		eqid 2763	. . . . . . . . . . . . . 14 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
75	1, 21, 74, 25, 24, 31	lspsnvs 21425	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LVec ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) = ((LSpan‘𝑊)‘{𝑥}))
76	70, 72, 73, 75	syl2an3an 1450	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) ∧ (𝐹 ∈ (LIndS‘𝑊) ∧ 𝑥 ∈ 𝐹)) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) = ((LSpan‘𝑊)‘{𝑥}))
77	76	an42s 674	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) = ((LSpan‘𝑊)‘{𝑥}))
78	77	sseq1d 3944	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
79	78	3adantl3 1187	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
80		eldifi 4070	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
81	19	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑊 ∈ LMod)
82	81	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) → 𝑊 ∈ LMod)
83		snssi 4746	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (Base‘𝑊) → {𝑋} ⊆ (Base‘𝑊))
84	2, 83, 6	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → (𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊))
85	84	ssdifssd 4086	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → ((𝐹 ∪ {𝑋}) ∖ {𝑥}) ⊆ (Base‘𝑊))
86	1, 37, 31	lspcl 21284	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ ((𝐹 ∪ {𝑋}) ∖ {𝑥}) ⊆ (Base‘𝑊)) → ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ∈ (LSubSp‘𝑊))
87	19, 85, 86	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LVec ∧ (𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊))) → ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ∈ (LSubSp‘𝑊))
88	87	3impb 1132	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ∈ (LSubSp‘𝑊))
89	88	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) → ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ∈ (LSubSp‘𝑊))
90	19	anim1i 627	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))))
91	1, 21, 74, 25	lmodvscl 21186	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
92	91	3expa 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
93	90, 73, 92	syl2an 608	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝐹 ∈ (LIndS‘𝑊) ∧ 𝑥 ∈ 𝐹)) → (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
94	93	an42s 674	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) → (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
95	94	3adantl3 1187	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) → (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
96	1, 37, 31, 82, 89, 95	ellspsn5b 21303	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) → ((𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
97	80, 96	sylanr2 696	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ((𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑥)}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
98	81	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ 𝐹) → 𝑊 ∈ LMod)
99	88	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ 𝐹) → ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ∈ (LSubSp‘𝑊))
100	73	3ad2antl2 1205	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (Base‘𝑊))
101	1, 37, 31, 98, 99, 100	ellspsn5b 21303	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
102	101	adantrr 730	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
103	79, 97, 102	3bitr4rd 315	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
104	3, 103	syl3anl3 1442	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
105	69, 104	bitrd 282	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑥 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑋})) ↔ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
106		difsnid 4771	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐹 → ((𝐹 ∖ {𝑥}) ∪ {𝑥}) = 𝐹)
107	106	fveq2d 6915	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐹 → ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑥})) = ((LSpan‘𝑊)‘𝐹))
108	107	eleq2d 2850	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 → (𝑋 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑥})) ↔ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
109	108	ad2antrl 741	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝑋 ∈ ((LSpan‘𝑊)‘((𝐹 ∖ {𝑥}) ∪ {𝑥})) ↔ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
110	40, 105, 109	3imtr3d 296	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ((𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) → 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
111	11, 110	mtod 201	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ (𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))
112	111	ralrimivva 3209	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))
113	10	adantr 486	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ¬ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹))
114		difsn 4761	. . . . . . . . . . 11 ⊢ (¬ 𝑋 ∈ 𝐹 → (𝐹 ∖ {𝑋}) = 𝐹)
115	54, 114	syl 18	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∖ {𝑋}) = 𝐹)
116	115	fveq2d 6915	. . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})) = ((LSpan‘𝑊)‘𝐹))
117	116	eleq2d 2850	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})) ↔ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘𝐹)))
118	117	adantr 486	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})) ↔ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘𝐹)))
119	1, 21, 74, 25, 24, 31	lspsnvs 21425	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))) ∧ 𝑋 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) = ((LSpan‘𝑊)‘{𝑋}))
120	119	3expa 1136	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊)))) ∧ 𝑋 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) = ((LSpan‘𝑊)‘{𝑋}))
121	120	an32s 665	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝑋 ∈ (Base‘𝑊)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊)))) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) = ((LSpan‘𝑊)‘{𝑋}))
122	71, 121	sylan2b 606	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) = ((LSpan‘𝑊)‘{𝑋}))
123	122	sseq1d 3944	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → (((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) ⊆ ((LSpan‘𝑊)‘𝐹) ↔ ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹)))
124	123	3adantl2 1186	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → (((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) ⊆ ((LSpan‘𝑊)‘𝐹) ↔ ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹)))
125	81	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑊 ∈ LMod)
126	1, 37, 31	lspcl 21284	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 ⊆ (Base‘𝑊)) → ((LSpan‘𝑊)‘𝐹) ∈ (LSubSp‘𝑊))
127	19, 2, 126	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) → ((LSpan‘𝑊)‘𝐹) ∈ (LSubSp‘𝑊))
128	127	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘𝐹) ∈ (LSubSp‘𝑊))
129	128	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((LSpan‘𝑊)‘𝐹) ∈ (LSubSp‘𝑊))
130	1, 21, 74, 25	lmodvscl 21186	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ (Base‘𝑊))
131	130	3expa 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ (Base‘𝑊))
132	131	an32s 665	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ (Base‘𝑊))
133	19, 132	sylanl1 693	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ (Base‘𝑊))
134	133	3adantl2 1186	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ (Base‘𝑊))
135	1, 37, 31, 125, 129, 134	ellspsn5b 21303	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘𝐹) ↔ ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) ⊆ ((LSpan‘𝑊)‘𝐹)))
136	80, 135	sylan2 605	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘𝐹) ↔ ((LSpan‘𝑊)‘{(𝑘( ·_𝑠 ‘𝑊)𝑋)}) ⊆ ((LSpan‘𝑊)‘𝐹)))
137		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋 ∈ (Base‘𝑊))
138	1, 37, 31, 81, 128, 137	ellspsn5b 21303	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) → (𝑋 ∈ ((LSpan‘𝑊)‘𝐹) ↔ ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹)))
139	138	adantr 486	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → (𝑋 ∈ ((LSpan‘𝑊)‘𝐹) ↔ ((LSpan‘𝑊)‘{𝑋}) ⊆ ((LSpan‘𝑊)‘𝐹)))
140	124, 136, 139	3bitr4d 314	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ (Base‘𝑊)) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘𝐹) ↔ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
141	3, 140	syl3anl3 1442	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘𝐹) ↔ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
142	118, 141	bitrd 282	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ((𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})) ↔ 𝑋 ∈ ((LSpan‘𝑊)‘𝐹)))
143	113, 142	mtbird 328	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))})) → ¬ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})))
144	143	ralrimiva 3158	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})))
145		oveq2 7454	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑘( ·_𝑠 ‘𝑊)𝑥) = (𝑘( ·_𝑠 ‘𝑊)𝑋))
146		sneq 4594	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
147	146	difeq2d 4066	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝐹 ∪ {𝑋}) ∖ {𝑥}) = ((𝐹 ∪ {𝑋}) ∖ {𝑋}))
148		difun2 4440	. . . . . . . . . . 11 ⊢ ((𝐹 ∪ {𝑋}) ∖ {𝑋}) = (𝐹 ∖ {𝑋})
149	147, 148	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐹 ∪ {𝑋}) ∖ {𝑥}) = (𝐹 ∖ {𝑋}))
150	149	fveq2d 6915	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) = ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋})))
151	145, 150	eleq12d 2858	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋}))))
152	151	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ¬ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋}))))
153	152	ralbidv 3189	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋}))))
154	153	ralsng 4636	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹)) → (∀𝑥 ∈ {𝑋}∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋}))))
155	154	3ad2ant3 1153	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (∀𝑥 ∈ {𝑋}∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑋) ∈ ((LSpan‘𝑊)‘(𝐹 ∖ {𝑋}))))
156	144, 155	mpbird 260	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ∀𝑥 ∈ {𝑋}∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))
157		ralunb 4198	. . 3 ⊢ (∀𝑥 ∈ (𝐹 ∪ {𝑋})∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ↔ (∀𝑥 ∈ 𝐹 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})) ∧ ∀𝑥 ∈ {𝑋}∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥}))))
158	112, 156, 157	sylanbrc 595	. 2 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ∀𝑥 ∈ (𝐹 ∪ {𝑋})∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))
159	1, 74, 31, 21, 25, 24	islinds2 22152	. . 3 ⊢ (𝑊 ∈ LVec → ((𝐹 ∪ {𝑋}) ∈ (LIndS‘𝑊) ↔ ((𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ (𝐹 ∪ {𝑋})∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))))
160	159	3ad2ant1 1151	. 2 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → ((𝐹 ∪ {𝑋}) ∈ (LIndS‘𝑊) ↔ ((𝐹 ∪ {𝑋}) ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ (𝐹 ∪ {𝑋})∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘((𝐹 ∪ {𝑋}) ∖ {𝑥})))))
161	8, 158, 160	mpbir2and 726	1 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∪ {𝑋}) ∈ (LIndS‘𝑊))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {csn 4588 ‘cfv 6561 (class class class)co 7446 Basecbs 17419 Scalarcsca 17463 ·_𝑠 cvsca 17464 0_gc0g 17642 Ringcrg 20492 NzRingcnzr 20795 LModclmod 21168 LSubSpclss 21239 LSpanclspn 21279 LVecclvec 21410 LIndSclinds 22144

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-tpos 8274 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-er 8748 df-en 9005 df-dom 9006 df-sdom 9007 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-2 12435 df-3 12436 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-0g 17644 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-grp 19179 df-minusg 19180 df-sbg 19181 df-cmn 20029 df-abl 20030 df-mgp 20394 df-rng 20408 df-ur 20441 df-ring 20494 df-oppr 20600 df-dvdsr 20620 df-unit 20621 df-invr 20651 df-nzr 20796 df-drng 21015 df-lmod 21170 df-lss 21240 df-lsp 21280 df-lvec 21411 df-lindf 22145 df-linds 22146

This theorem is used by: (None)

W3C validator