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Theorem lsspropdg 13708

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 3171	. . . . . . . . . . . . . 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 2572	. . . . . . . . . . . . . . 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 5917	. . . . . . . . . . . . . 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 1248	. . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑏 ∈ 𝐵)
15	12, 14	sseldd 3171	. . . . . . . . . . . . 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 5918	. . . . . . . . . . . . 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 1247	. . . . . . . . . . . 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 5918	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑎 ∈ 𝐵)) → (𝑧( ·_𝑠 ‘𝐾)𝑎) = (𝑧( ·_𝑠 ‘𝐿)𝑎))
21	1, 2, 5, 20	syl12anc 1247	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → (𝑧( ·_𝑠 ‘𝐾)𝑎) = (𝑧( ·_𝑠 ‘𝐿)𝑎))
22	21	oveq1d 5906	. . . . . . . . . . . 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 2222	. . . . . . . . . . 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 2258	. . . . . . . . . 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 2512	. . . . . . . 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 2486	. . . . . . 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 984	. . . . 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 984	. . . . 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 3200	. . . . 5 ⊢ (𝜑 → (𝑠 ⊆ 𝐵 ↔ 𝑠 ⊆ (Base‘𝐾)))
35		lsspropd.p1	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
36	35	raleqdv 2692	. . . . 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 1325	. . . 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 3200	. . . . 5 ⊢ (𝜑 → (𝑠 ⊆ 𝐵 ↔ 𝑠 ⊆ (Base‘𝐿)))
40		lsspropd.p2	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
41	40	raleqdv 2692	. . . . 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 1325	. . . 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 2189	. . . . 5 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
46		eqid 2189	. . . . 5 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
47		eqid 2189	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
48		eqid 2189	. . . . 5 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
49		eqid 2189	. . . . 5 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
50		eqid 2189	. . . . 5 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾)
51	45, 46, 47, 48, 49, 50	islssmg 13635	. . . 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 2189	. . . . 5 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
55		eqid 2189	. . . . 5 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
56		eqid 2189	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
57		eqid 2189	. . . . 5 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
58		eqid 2189	. . . . 5 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
59		eqid 2189	. . . . 5 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
60	54, 55, 56, 57, 58, 59	islssmg 13635	. . . 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 2187	1 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 ⊆ wss 3144 ‘cfv 5231 (class class class)co 5891 Basecbs 12480 +_gcplusg 12555 Scalarcsca 12558 ·_𝑠 cvsca 12559 LSubSpclss 13629

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-ov 5894 df-inn 8938 df-ndx 12483 df-slot 12484 df-base 12486 df-lssm 13630

This theorem is referenced by: lsppropd 13709 lidlrsppropdg 13772

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