Theorem lsspropd 20194

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)))

Assertion

Ref	Expression
lsspropd	⊢ (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

Proof of Theorem lsspropd

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

Step	Hyp	Ref	Expression
1		simpll 763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝜑)
2		simprl 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑧 ∈ 𝑃)
3		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑠 ⊆ 𝐵)
4		simprrl 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑎 ∈ 𝑠)
5	3, 4	sseldd 3918	. . . . . . . . . . . . . 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 3114	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
8	7	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
9		ovrspc2v 7281	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝐵 ⊆ 𝑊)
13		simprrr 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑏 ∈ 𝑠)
14	3, 13	sseldd 3918	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → 𝑏 ∈ 𝐵)
15	12, 14	sseldd 3918	. . . . . . . . . . . . 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 7282	. . . . . . . . . . . . 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 833	. . . . . . . . . . . 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 7282	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ 𝑎 ∈ 𝐵)) → (𝑧( ·𝑠 ‘𝐾)𝑎) = (𝑧( ·𝑠 ‘𝐿)𝑎))
21	1, 2, 5, 20	syl12anc 833	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ⊆ 𝐵) ∧ (𝑧 ∈ 𝑃 ∧ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ 𝑠))) → (𝑧( ·𝑠 ‘𝐾)𝑎) = (𝑧( ·𝑠 ‘𝐿)𝑎))
22	21	oveq1d 7270	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 2823	. . . . . . . . . 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 467	. . . . . . . . 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 3121	. . . . . . . 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 3119	. . . . . . 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 628	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → ((𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
29	28	pm5.32da 578	. . . . 5 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ (𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)) ↔ (𝑠 ⊆ 𝐵 ∧ (𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))))
30		3anass 1093	. . . . 5 ⊢ ((𝑠 ⊆ 𝐵 ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ (𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠)))
31		3anass 1093	. . . . 5 ⊢ ((𝑠 ⊆ 𝐵 ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ (𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
32	29, 30, 31	3bitr4g 313	. . . 4 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ 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 3949	. . . . 5 ⊢ (𝜑 → (𝑠 ⊆ 𝐵 ↔ 𝑠 ⊆ (Base‘𝐾)))
35		lsspropd.p1	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
36	35	raleqdv 3339	. . . . 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 1436	. . . 4 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐾) ∧ 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 3949	. . . . 5 ⊢ (𝜑 → (𝑠 ⊆ 𝐵 ↔ 𝑠 ⊆ (Base‘𝐿)))
40		lsspropd.p2	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
41	40	raleqdv 3339	. . . . 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 1436	. . . 4 ⊢ (𝜑 → ((𝑠 ⊆ 𝐵 ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ 𝑃 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
43	32, 37, 42	3bitr3d 308	. . 3 ⊢ (𝜑 → ((𝑠 ⊆ (Base‘𝐾) ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠)))
44		eqid 2738	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
45		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
46		eqid 2738	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
47		eqid 2738	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
48		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
49		eqid 2738	. . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾)
50	44, 45, 46, 47, 48, 49	islss 20111	. . 3 ⊢ (𝑠 ∈ (LSubSp‘𝐾) ↔ (𝑠 ⊆ (Base‘𝐾) ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐾))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐾)𝑎)(+g‘𝐾)𝑏) ∈ 𝑠))
51		eqid 2738	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
52		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
53		eqid 2738	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
54		eqid 2738	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
55		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
56		eqid 2738	. . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
57	51, 52, 53, 54, 55, 56	islss 20111	. . 3 ⊢ (𝑠 ∈ (LSubSp‘𝐿) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ 𝑠 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑧( ·𝑠 ‘𝐿)𝑎)(+g‘𝐿)𝑏) ∈ 𝑠))
58	43, 50, 57	3bitr4g 313	. 2 ⊢ (𝜑 → (𝑠 ∈ (LSubSp‘𝐾) ↔ 𝑠 ∈ (LSubSp‘𝐿)))
59	58	eqrdv 2736	1 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ⊆ wss 3883  ∅c0 4253  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891   ·_𝑠 cvsca 16892  LSubSpclss 20108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-lss 20109

This theorem is referenced by:  lsppropd  20195  lidlrsppropd  20414  ply1lss  21277