Theorem sspph 27094

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 27053	. . 3 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
2		sspph.h	. . . 4 ⊢ 𝐻 = (SubSp‘𝑈)
3	2	sspnv 26965	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
4	1, 3	sylan 487	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
5		eqid 2610	. . . . . . . . . 10 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2610	. . . . . . . . . 10 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
7	5, 6, 2	sspba 26966	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (BaseSet‘𝑊) ⊆ (BaseSet‘𝑈))
8	7	sseld 3567	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ (BaseSet‘𝑊) → 𝑥 ∈ (BaseSet‘𝑈)))
9	7	sseld 3567	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ (BaseSet‘𝑊) → 𝑦 ∈ (BaseSet‘𝑈)))
10	8, 9	anim12d 584	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))))
11	1, 10	sylan 487	. . . . . 6 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))))
12	11	imp 444	. . . . 5 ⊢ (((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈)))
13		eqid 2610	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
14		eqid 2610	. . . . . . . 8 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈)
15		eqid 2610	. . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
16	5, 13, 14, 15	phpar2 27062	. . . . . . 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 1258	. . . . . 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 747	. . . . 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 486	. . . 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 2610	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
21	6, 20	nvgcl 26859	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
22	21	3expb 1258	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
23	3, 22	sylan 487	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
24		eqid 2610	. . . . . . . . . . 11 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
25	6, 15, 24, 2	sspnval 26976	. . . . . . . . . 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 1257	. . . . . . . . 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 486	. . . . . . . 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 6564	. . . . . . 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 26968	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣 ‘𝑊)𝑦) = (𝑥( +𝑣 ‘𝑈)𝑦))
30	29	fveq2d 6107	. . . . . . . 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 6564	. . . . . . 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 2644	. . . . . 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 2610	. . . . . . . . . . . 12 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊)
34	6, 33	nvmcl 26885	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
35	34	3expb 1258	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
36	3, 35	sylan 487	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣 ‘𝑊)𝑦) ∈ (BaseSet‘𝑊))
37	6, 15, 24, 2	sspnval 26976	. . . . . . . . . 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 1257	. . . . . . . . 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 486	. . . . . . . 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 26972	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣 ‘𝑊)𝑦) = (𝑥( −𝑣 ‘𝑈)𝑦))
41	40	fveq2d 6107	. . . . . . . 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 2644	. . . . . . 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 6564	. . . . . 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 6567	. . . . 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 680	. . . 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 26976	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝑥 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑥) = ((normCV‘𝑈)‘𝑥))
47	46	3expa 1257	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑥) = ((normCV‘𝑈)‘𝑥))
48	47	adantrr 749	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘𝑥) = ((normCV‘𝑈)‘𝑥))
49	48	oveq1d 6564	. . . . . . 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 26976	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑦) = ((normCV‘𝑈)‘𝑦))
51	50	3expa 1257	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((normCV‘𝑊)‘𝑦) = ((normCV‘𝑈)‘𝑦))
52	51	adantrl 748	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV‘𝑊)‘𝑦) = ((normCV‘𝑈)‘𝑦))
53	52	oveq1d 6564	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV‘𝑊)‘𝑦)↑2) = (((normCV‘𝑈)‘𝑦)↑2))
54	49, 53	oveq12d 6567	. . . . . 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 680	. . . . 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 6565	. . . 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 2654	. . 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 2954	. 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 27061	. 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 695	1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ CPreHilOLD)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549   + caddc 9818   · cmul 9820  2c2 10947  ↑cexp 12722  NrmCVeccnv 26823   +_𝑣 cpv 26824  BaseSetcba 26825   −_𝑣 cnsb 26828  norm_CVcnmcv 26829  SubSpcss 26960  CPreHil_OLDccphlo 27051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-sub 10147  df-neg 10148  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ssp 26961  df-ph 27052

This theorem is referenced by:  ssphl  27157  hhssph  27515