Theorem lbspropd 21407

Description: If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)

Hypotheses

Ref	Expression
lbspropd.b1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lbspropd.b2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
lbspropd.w	⊢ (𝜑 → 𝐵 ⊆ 𝑊)
lbspropd.p	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
lbspropd.s1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
lbspropd.s2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
lbspropd.f	⊢ 𝐹 = (Scalar‘𝐾)
lbspropd.g	⊢ 𝐺 = (Scalar‘𝐿)
lbspropd.p1	⊢ (𝜑 → 𝑃 = (Base‘𝐹))
lbspropd.p2	⊢ (𝜑 → 𝑃 = (Base‘𝐺))
lbspropd.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))
lbspropd.v1	⊢ (𝜑 → 𝐾 ∈ 𝑋)
lbspropd.v2	⊢ (𝜑 → 𝐿 ∈ 𝑌)

Assertion

Ref	Expression
lbspropd	⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝑃,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem lbspropd

Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 787	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → 𝜑)
2		eldifi 4070	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)}) → 𝑣 ∈ 𝑃)
3	2	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → 𝑣 ∈ 𝑃)
4		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ 𝐵)
5	4	sselda 3913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → 𝑢 ∈ 𝐵)
6	5	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → 𝑢 ∈ 𝐵)
7		lbspropd.s2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
8	7	oveqrspc2v 7473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑃 ∧ 𝑢 ∈ 𝐵)) → (𝑣( ·𝑠 ‘𝐾)𝑢) = (𝑣( ·𝑠 ‘𝐿)𝑢))
9	1, 3, 6, 8	syl12anc 850	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → (𝑣( ·𝑠 ‘𝐾)𝑢) = (𝑣( ·𝑠 ‘𝐿)𝑢))
10		lbspropd.b1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
11		lbspropd.b2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
12		lbspropd.w	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝑊)
13		lbspropd.p	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
14		lbspropd.s1	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
15		lbspropd.p1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 = (Base‘𝐹))
16		lbspropd.f	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = (Scalar‘𝐾)
17	16	fveq2i 6914	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘(Scalar‘𝐾))
18	15, 17	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
19		lbspropd.p2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 = (Base‘𝐺))
20		lbspropd.g	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (Scalar‘𝐿)
21	20	fveq2i 6914	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘(Scalar‘𝐿))
22	19, 21	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
23		lbspropd.v1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
24		lbspropd.v2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐿 ∈ 𝑌)
25	10, 11, 12, 13, 14, 7, 18, 22, 23, 24	lsppropd 21326	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
26	1, 25	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → (LSpan‘𝐾) = (LSpan‘𝐿))
27	26	fveq1d 6913	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) = ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))
28	9, 27	eleq12d 2858	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → ((𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
29	28	notbid 321	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → (¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
30	29	ralbidva 3187	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
31	15	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → 𝑃 = (Base‘𝐹))
32	31	difeq1d 4065	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (𝑃 ∖ {(0g‘𝐹)}) = ((Base‘𝐹) ∖ {(0g‘𝐹)}))
33	32	raleqdv 3323	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))))
34	19	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → 𝑃 = (Base‘𝐺))
35		lbspropd.a	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))
36	15, 19, 35	grpidpropd 18874	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐺))
37	36	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (0g‘𝐹) = (0g‘𝐺))
38	37	sneqd 4596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → {(0g‘𝐹)} = {(0g‘𝐺)})
39	34, 38	difeq12d 4067	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (𝑃 ∖ {(0g‘𝐹)}) = ((Base‘𝐺) ∖ {(0g‘𝐺)}))
40	39	raleqdv 3323	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
41	30, 33, 40	3bitr3d 312	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
42	41	ralbidva 3187	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
43	42	anbi2d 642	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → ((((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))) ↔ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
44	43	pm5.32da 590	. . . . 5 ⊢ (𝜑 → ((𝑧 ⊆ 𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ 𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))))
45	10	sseq2d 3945	. . . . . 6 ⊢ (𝜑 → (𝑧 ⊆ 𝐵 ↔ 𝑧 ⊆ (Base‘𝐾)))
46	45	anbi1d 643	. . . . 5 ⊢ (𝜑 → ((𝑧 ⊆ 𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))))))
47	11	sseq2d 3945	. . . . . 6 ⊢ (𝜑 → (𝑧 ⊆ 𝐵 ↔ 𝑧 ⊆ (Base‘𝐿)))
48	25	fveq1d 6913	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘𝐾)‘𝑧) = ((LSpan‘𝐿)‘𝑧))
49	10, 11	eqtr3d 2800	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
50	48, 49	eqeq12d 2779	. . . . . . 7 ⊢ (𝜑 → (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ↔ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿)))
51	50	anbi1d 643	. . . . . 6 ⊢ (𝜑 → ((((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))) ↔ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
52	47, 51	anbi12d 644	. . . . 5 ⊢ (𝜑 → ((𝑧 ⊆ 𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))))
53	44, 46, 52	3bitr3d 312	. . . 4 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))))
54		3anass 1111	. . . 4 ⊢ ((𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))))
55		3anass 1111	. . . 4 ⊢ ((𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
56	53, 54, 55	3bitr4g 317	. . 3 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
57		eqid 2763	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
58		eqid 2763	. . . . 5 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
59		eqid 2763	. . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹)
60		eqid 2763	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
61		eqid 2763	. . . . 5 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
62		eqid 2763	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
63	57, 16, 58, 59, 60, 61, 62	islbs 21384	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (𝑧 ∈ (LBasis‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))))
64	23, 63	syl 18	. . 3 ⊢ (𝜑 → (𝑧 ∈ (LBasis‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g‘𝐹)}) ¬ (𝑣( ·𝑠 ‘𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))))
65		eqid 2763	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
66		eqid 2763	. . . . 5 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
67		eqid 2763	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
68		eqid 2763	. . . . 5 ⊢ (LBasis‘𝐿) = (LBasis‘𝐿)
69		eqid 2763	. . . . 5 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
70		eqid 2763	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
71	65, 20, 66, 67, 68, 69, 70	islbs 21384	. . . 4 ⊢ (𝐿 ∈ 𝑌 → (𝑧 ∈ (LBasis‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
72	24, 71	syl 18	. . 3 ⊢ (𝜑 → (𝑧 ∈ (LBasis‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g‘𝐺)}) ¬ (𝑣( ·𝑠 ‘𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
73	56, 64, 72	3bitr4d 314	. 2 ⊢ (𝜑 → (𝑧 ∈ (LBasis‘𝐾) ↔ 𝑧 ∈ (LBasis‘𝐿)))
74	73	eqrdv 2761	1 ⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  +_gcplusg 17460  Scalarcsca 17463   ·_𝑠 cvsca 17464  0_gc0g 17642  LSpanclspn 21279  LBasisclbs 21382

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  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-id 5551  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-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-0g 17644  df-lss 21240  df-lsp 21280  df-lbs 21383

This theorem is used by:  dimpropd  34324