Theorem sspph 27710

Description: A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sspph.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
sspph	⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ CPreHilOLD)

Proof of Theorem sspph

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phnv 27669	. . 3 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
2		sspph.h	. . . 4 ⊢ 𝐻 = (SubSp‘𝑈)
3	2	sspnv 27581	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
4	1, 3	sylan 488	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
5		eqid 2622	. . . . . . . . . 10 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2622	. . . . . . . . . 10 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
7	5, 6, 2	sspba 27582	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (BaseSet‘𝑊) ⊆ (BaseSet‘𝑈))
8	7	sseld 3602	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ (BaseSet‘𝑊) → 𝑥 ∈ (BaseSet‘𝑈)))
9	7	sseld 3602	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ (BaseSet‘𝑊) → 𝑦 ∈ (BaseSet‘𝑈)))
10	8, 9	anim12d 586	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))))
11	1, 10	sylan 488	. . . . . 6 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))))
12	11	imp 445	. . . . 5 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈)))
13		eqid 2622	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
14		eqid 2622	. . . . . . . 8 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈)
15		eqid 2622	. . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
16	5, 13, 14, 15	phpar2 27678	. . . . . . 7 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2) + (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2)) = (2 · ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2))))
17	16	3expb 1266	. . . . . 6 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))) → ((((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2) + (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2)) = (2 · ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2))))
18	17	adantlr 751	. . . . 5 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))) → ((((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2) + (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2)) = (2 · ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2))))
19	12, 18	syldan 487	. . . 4 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2) + (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2)) = (2 · ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2))))
20		eqid 2622	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
21	6, 20	nvgcl 27475	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
22	21	3expb 1266	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
23	3, 22	sylan 488	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
24		eqid 2622	. . . . . . . . . . 11 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
25	6, 15, 24, 2	sspnval 27592	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑊)𝑦)))
26	25	3expa 1265	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑊)𝑦)))
27	23, 26	syldan 487	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑊)𝑦)))
28	27	oveq1d 6665	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) = (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2))
29	6, 13, 20, 2	sspgval 27584	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣 ‘𝑊)𝑦) = (𝑥( +𝑣 ‘𝑈)𝑦))
30	29	fveq2d 6195	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦)))
31	30	oveq1d 6665	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) = (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2))
32	28, 31	eqtrd 2656	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) = (((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2))
33		eqid 2622	. . . . . . . . . . . 12 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊)
34	6, 33	nvmcl 27501	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
35	34	3expb 1266	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
36	3, 35	sylan 488	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
37	6, 15, 24, 2	sspnval 27592	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑊)𝑦)))
38	37	3expa 1265	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑊)𝑦)))
39	36, 38	syldan 487	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑊)𝑦)))
40	6, 14, 33, 2	sspmval 27588	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣 ‘𝑊)𝑦) = (𝑥( −𝑣 ‘𝑈)𝑦))
41	40	fveq2d 6195	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦)))
42	39, 41	eqtrd 2656	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦)) = ((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦)))
43	42	oveq1d 6665	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦))↑2) = (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2))
44	32, 43	oveq12d 6668	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) + (((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦))↑2)) = ((((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2) + (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2)))
45	1, 44	sylanl1 682	. . . 4 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) + (((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦))↑2)) = ((((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦))↑2) + (((normCV‘𝑈)‘(𝑥( −𝑣 ‘𝑈)𝑦))↑2)))
46	6, 15, 24, 2	sspnval 27592	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝑥 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑥) = ((normCV‘𝑈)‘𝑥))
47	46	3expa 1265	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑥) = ((normCV‘𝑈)‘𝑥))
48	47	adantrr 753	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘𝑥) = ((normCV‘𝑈)‘𝑥))
49	48	oveq1d 6665	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑊)‘𝑥)↑2) = (((normCV‘𝑈)‘𝑥)↑2))
50	6, 15, 24, 2	sspnval 27592	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑦) = ((normCV‘𝑈)‘𝑦))
51	50	3expa 1265	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑦) = ((normCV‘𝑈)‘𝑦))
52	51	adantrl 752	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘𝑦) = ((normCV‘𝑈)‘𝑦))
53	52	oveq1d 6665	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑊)‘𝑦)↑2) = (((normCV‘𝑈)‘𝑦)↑2))
54	49, 53	oveq12d 6668	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV‘𝑊)‘𝑥)↑2) + (((normCV‘𝑊)‘𝑦)↑2)) = ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2)))
55	1, 54	sylanl1 682	. . . . 5 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV‘𝑊)‘𝑥)↑2) + (((normCV‘𝑊)‘𝑦)↑2)) = ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2)))
56	55	oveq2d 6666	. . . 4 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (2 · ((((normCV‘𝑊)‘𝑥)↑2) + (((normCV‘𝑊)‘𝑦)↑2))) = (2 · ((((normCV‘𝑈)‘𝑥)↑2) + (((normCV‘𝑈)‘𝑦)↑2))))
57	19, 45, 56	3eqtr4d 2666	. . 3 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) + (((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦))↑2)) = (2 · ((((normCV‘𝑊)‘𝑥)↑2) + (((normCV‘𝑊)‘𝑦)↑2))))
58	57	ralrimivva 2971	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ (BaseSet‘𝑊)∀𝑦 ∈ (BaseSet‘𝑊)((((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) + (((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦))↑2)) = (2 · ((((normCV‘𝑊)‘𝑥)↑2) + (((normCV‘𝑊)‘𝑦)↑2))))
59	6, 20, 33, 24	isph 27677	. 2 ⊢ (𝑊 ∈ CPreHilOLD ↔ (𝑊 ∈ NrmCVec ∧ ∀𝑥 ∈ (BaseSet‘𝑊)∀𝑦 ∈ (BaseSet‘𝑊)((((normCV‘𝑊)‘(𝑥( +𝑣 ‘𝑊)𝑦))↑2) + (((normCV‘𝑊)‘(𝑥( −𝑣 ‘𝑊)𝑦))↑2)) = (2 · ((((normCV‘𝑊)‘𝑥)↑2) + (((normCV‘𝑊)‘𝑦)↑2)))))
60	4, 58, 59	sylanbrc 698	1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ CPreHilOLD)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650   + caddc 9939   · cmul 9941  2c2 11070  ↑cexp 12860  NrmCVeccnv 27439   +_𝑣 cpv 27440  BaseSetcba 27441   −_𝑣 cnsb 27444  norm_CVcnmcv 27445  SubSpcss 27576  CPreHil_OLDccphlo 27667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ssp 27577  df-ph 27668

This theorem is referenced by:  ssphl  27773  hhssph  28131