Theorem lsspropdg 14389

Description: If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)

Hypotheses

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

Assertion

Ref	Expression
lsspropdg	⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))

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

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

Proof of Theorem lsspropdg

Dummy variables 𝑎 𝑏 𝑧 𝑠 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝜑)
2		simprl 529	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑧 ∈ 𝑃)
3		simplr 528	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑠 ⊆ 𝐵)
4		simprrl 539	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑎 ∈ 𝑠)
5	3, 4	sseldd 3225	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑎 ∈ 𝐵)
6		lsspropd.s1	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
7	6	ralrimivva 2612	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
8	7	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
9		ovrspc2v 6026	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝑃 ∧ 𝑎 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) → (𝑧( ·𝑠 ‘𝐾)𝑎) ∈ 𝑊)
10	2, 5, 8, 9	syl21anc 1270	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → (𝑧( ·𝑠 ‘𝐾)𝑎) ∈ 𝑊)
11		lsspropd.w	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝑊)
12	11	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝐵 ⊆ 𝑊)
13		simprrr 540	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑏 ∈ 𝑠)
14	3, 13	sseldd 3225	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑏 ∈ 𝐵)
15	12, 14	sseldd 3225	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑏 ∈ 𝑊)
16		lsspropd.p	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
17	16	oveqrspc2v 6027	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑧( ·𝑠 ‘𝐾)𝑎) ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) = ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐿)𝑏))
18	1, 10, 15, 17	syl12anc 1269	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) = ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐿)𝑏))
19		lsspropd.s2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
20	19	oveqrspc2v 6027	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑎 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐾)𝑎) = (𝑧( ·𝑠 ‘𝐿)𝑎))
21	1, 2, 5, 20	syl12anc 1269	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → (𝑧( ·𝑠 ‘𝐾)𝑎) = (𝑧( ·𝑠 ‘𝐿)𝑎))
22	21	oveq1d 6015	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐿)𝑏) = ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏))
23	18, 22	eqtrd 2262	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) = ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏))
24	23	eleq1d 2298	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → (((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠 ↔ ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))
25	24	anassrs 400	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑃) ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠)) → (((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠 ↔ ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))
26	25	2ralbidva 2552	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑃) → (∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠 ↔ ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))
27	26	ralbidva 2526	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → (∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠 ↔ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))
28	27	anbi2d 464	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → ((∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
29	28	pm5.32da 452	. . . . 5 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)) ↔ (𝑠 ⊆ 𝐵 ∧ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))))
30		3anass 1006	. . . . 5 ⊢ ((𝑠 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)))
31		3anass 1006	. . . . 5 ⊢ ((𝑠 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
32	29, 30, 31	3bitr4g 223	. . . 4 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
33		lsspropd.b1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
34	33	sseq2d 3254	. . . . 5 ⊢ (𝜑 → (𝑠 ⊆ 𝐵 ↔ 𝑠 ⊆ (Base‘𝐾)))
35		lsspropd.p1	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
36	35	raleqdv 2734	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠 ↔ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠))
37	34, 36	3anbi13d 1348	. . . 4 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐾) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)))
38		lsspropd.b2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
39	38	sseq2d 3254	. . . . 5 ⊢ (𝜑 → (𝑠 ⊆ 𝐵 ↔ 𝑠 ⊆ (Base‘𝐿)))
40		lsspropd.p2	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
41	40	raleqdv 2734	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠 ↔ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))
42	39, 41	3anbi13d 1348	. . . 4 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
43	32, 37, 42	3bitr3d 218	. . 3 ⊢ (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
44		lsppropd.v1	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
45		eqid 2229	. . . . 5 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
46		eqid 2229	. . . . 5 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
47		eqid 2229	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
48		eqid 2229	. . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐾)
49		eqid 2229	. . . . 5 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
50		eqid 2229	. . . . 5 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾)
51	45, 46, 47, 48, 49, 50	islssmg 14316	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (𝑠 ∈ (LSubSp‘𝐾) ↔ (𝑠 ⊆ (Base‘𝐾) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)))
52	44, 51	syl 14	. . 3 ⊢ (𝜑 → (𝑠 ∈ (LSubSp‘𝐾) ↔ (𝑠 ⊆ (Base‘𝐾) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)))
53		lsppropd.v2	. . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝑌)
54		eqid 2229	. . . . 5 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
55		eqid 2229	. . . . 5 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
56		eqid 2229	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
57		eqid 2229	. . . . 5 ⊢ (+g‘𝐿) = (+g‘𝐿)
58		eqid 2229	. . . . 5 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
59		eqid 2229	. . . . 5 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
60	54, 55, 56, 57, 58, 59	islssmg 14316	. . . 4 ⊢ (𝐿 ∈ 𝑌 → (𝑠 ∈ (LSubSp‘𝐿) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
61	53, 60	syl 14	. . 3 ⊢ (𝜑 → (𝑠 ∈ (LSubSp‘𝐿) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ ∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
62	43, 52, 61	3bitr4d 220	. 2 ⊢ (𝜑 → (𝑠 ∈ (LSubSp‘𝐾) ↔ 𝑠 ∈ (LSubSp‘𝐿)))
63	62	eqrdv 2227	1 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  Scalarcsca 13108   ·_𝑠 cvsca 13109  LSubSpclss 14310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-lssm 14311

This theorem is referenced by:  lsppropd  14390  lidlrsppropdg  14453