Theorem lbspropd 21121

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 774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → 𝜑)
2		eldifi 4154	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)}) → 𝑣 ∈ 𝑃)
3	2	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → 𝑣 ∈ 𝑃)
4		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ 𝐵)
5	4	sselda 4008	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → 𝑢 ∈ 𝐵)
6	5	adantr 480	. . . . . . . . . . . . 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 7475	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑃 ∧ 𝑢 ∈ 𝐵)) → (𝑣( ·𝑠 ‘𝐾)𝑢) = (𝑣( ·𝑠 ‘𝐿)𝑢))
9	1, 3, 6, 8	syl12anc 836	. . . . . . . . . . . 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 6923	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘(Scalar‘𝐾))
18	15, 17	eqtrdi 2796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
19		lbspropd.p2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 = (Base‘𝐺))
20		lbspropd.g	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (Scalar‘𝐿)
21	20	fveq2i 6923	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘(Scalar‘𝐿))
22	19, 21	eqtrdi 2796	. . . . . . . . . . . . . . 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 21040	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
26	1, 25	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → (LSpan‘𝐾) = (LSpan‘𝐿))
27	26	fveq1d 6922	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g‘𝐹)})) → ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) = ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))
28	9, 27	eleq12d 2838	. . . . . . . . . . 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 318	. . . . . . . . . 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 3182	. . . . . . . . 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 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → 𝑃 = (Base‘𝐹))
32	31	difeq1d 4148	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (𝑃 ∖ {(0g‘𝐹)}) = ((Base‘𝐹) ∖ {(0g‘𝐹)}))
33	32	raleqdv 3334	. . . . . . . . 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 725	. . . . . . . . . . 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 18700	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐺))
37	36	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (0g‘𝐹) = (0g‘𝐺))
38	37	sneqd 4660	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → {(0g‘𝐹)} = {(0g‘𝐺)})
39	34, 38	difeq12d 4150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ 𝐵) ∧ 𝑢 ∈ 𝑧) → (𝑃 ∖ {(0g‘𝐹)}) = ((Base‘𝐺) ∖ {(0g‘𝐺)}))
40	39	raleqdv 3334	. . . . . . . . 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 309	. . . . . . . 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 3182	. . . . . . 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 629	. . . . . 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 578	. . . . 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 4041	. . . . . 6 ⊢ (𝜑 → (𝑧 ⊆ 𝐵 ↔ 𝑧 ⊆ (Base‘𝐾)))
46	45	anbi1d 630	. . . . 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 4041	. . . . . 6 ⊢ (𝜑 → (𝑧 ⊆ 𝐵 ↔ 𝑧 ⊆ (Base‘𝐿)))
48	25	fveq1d 6922	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘𝐾)‘𝑧) = ((LSpan‘𝐿)‘𝑧))
49	10, 11	eqtr3d 2782	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
50	48, 49	eqeq12d 2756	. . . . . . 7 ⊢ (𝜑 → (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ↔ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿)))
51	50	anbi1d 630	. . . . . 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 631	. . . . 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 309	. . . 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 1095	. . . 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 1095	. . . 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 314	. . 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 2740	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
58		eqid 2740	. . . . 5 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
59		eqid 2740	. . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹)
60		eqid 2740	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
61		eqid 2740	. . . . 5 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
62		eqid 2740	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
63	57, 16, 58, 59, 60, 61, 62	islbs 21098	. . . 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 17	. . 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 2740	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
66		eqid 2740	. . . . 5 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
67		eqid 2740	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
68		eqid 2740	. . . . 5 ⊢ (LBasis‘𝐿) = (LBasis‘𝐿)
69		eqid 2740	. . . . 5 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
70		eqid 2740	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
71	65, 20, 66, 67, 68, 69, 70	islbs 21098	. . . 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 17	. . 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 311	. 2 ⊢ (𝜑 → (𝑧 ∈ (LBasis‘𝐾) ↔ 𝑧 ∈ (LBasis‘𝐿)))
74	73	eqrdv 2738	1 ⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  LSpanclspn 20992  LBasisclbs 21096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-0g 17501  df-lss 20953  df-lsp 20993  df-lbs 21097

This theorem is referenced by:  dimpropd  33621