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Theorem hhssnv 31350

Description: Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hhssnvt.1	⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
hhssnv.2	⊢ 𝐻 ∈ S_ℋ

**Assertion**
Ref	Expression
hhssnv	⊢ 𝑊 ∈ NrmCVec

Proof of Theorem hhssnv Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hhssnv.2	. . . . 5 ⊢ 𝐻 ∈ S_ℋ
2	1	hhssabloi 31348	. . . 4 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp
3		ablogrpo 30633	. . . 4 ⊢ (( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp)
4	2, 3	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp
5	1	shssii 31299	. . . . . 6 ⊢ 𝐻 ⊆ ℋ
6		xpss12 5639	. . . . . 6 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	5, 5, 6	mp2an 693	. . . . 5 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)
8		ax-hfvadd 31086	. . . . . 6 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
9	8	fdmi 6673	. . . . 5 ⊢ dom +_ℎ = ( ℋ × ℋ)
10	7, 9	sseqtrri 3972	. . . 4 ⊢ (𝐻 × 𝐻) ⊆ dom +_ℎ
11		ssdmres 5972	. . . 4 ⊢ ((𝐻 × 𝐻) ⊆ dom +_ℎ ↔ dom ( +_ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻))
12	10, 11	mpbi 230	. . 3 ⊢ dom ( +_ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻)
13	4, 12	grporn 30607	. 2 ⊢ 𝐻 = ran ( +_ℎ ↾ (𝐻 × 𝐻))
14		sh0 31302	. . . . . 6 ⊢ (𝐻 ∈ S_ℋ → 0_ℎ ∈ 𝐻)
15	1, 14	ax-mp 5	. . . . 5 ⊢ 0_ℎ ∈ 𝐻
16		ovres 7526	. . . . 5 ⊢ ((0_ℎ ∈ 𝐻 ∧ 0_ℎ ∈ 𝐻) → (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = (0_ℎ +_ℎ 0_ℎ))
17	15, 15, 16	mp2an 693	. . . 4 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = (0_ℎ +_ℎ 0_ℎ)
18		ax-hv0cl 31089	. . . . 5 ⊢ 0_ℎ ∈ ℋ
19	18	hvaddlidi 31115	. . . 4 ⊢ (0_ℎ +_ℎ 0_ℎ) = 0_ℎ
20	17, 19	eqtri 2760	. . 3 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ
21		eqid 2737	. . . . 5 ⊢ (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) = (GId‘( +_ℎ ↾ (𝐻 × 𝐻)))
22	13, 21	grpoid 30606	. . . 4 ⊢ ((( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ 0_ℎ ∈ 𝐻) → (0_ℎ = (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) ↔ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ))
23	4, 15, 22	mp2an 693	. . 3 ⊢ (0_ℎ = (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) ↔ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ)
24	20, 23	mpbir 231	. 2 ⊢ 0_ℎ = (GId‘( +_ℎ ↾ (𝐻 × 𝐻)))
25		ax-hfvmul 31091	. . . . . 6 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
26		ffn 6662	. . . . . 6 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
27	25, 26	ax-mp 5	. . . . 5 ⊢ ·_ℎ Fn (ℂ × ℋ)
28		ssid 3945	. . . . . 6 ⊢ ℂ ⊆ ℂ
29		xpss12 5639	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐻 ⊆ ℋ) → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
30	28, 5, 29	mp2an 693	. . . . 5 ⊢ (ℂ × 𝐻) ⊆ (ℂ × ℋ)
31		fnssres 6615	. . . . 5 ⊢ (( ·_ℎ Fn (ℂ × ℋ) ∧ (ℂ × 𝐻) ⊆ (ℂ × ℋ)) → ( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻))
32	27, 30, 31	mp2an 693	. . . 4 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻)
33		ovelrn 7536	. . . . . . 7 ⊢ (( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻) → (𝑧 ∈ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ↔ ∃𝑥 ∈ ℂ ∃𝑦 ∈ 𝐻 𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦)))
34	32, 33	ax-mp 5	. . . . . 6 ⊢ (𝑧 ∈ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ↔ ∃𝑥 ∈ ℂ ∃𝑦 ∈ 𝐻 𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦))
35		ovres 7526	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) = (𝑥 ·_ℎ 𝑦))
36		shmulcl 31304	. . . . . . . . . 10 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
37	1, 36	mp3an1 1451	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
38	35, 37	eqeltrd 2837	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) ∈ 𝐻)
39		eleq1 2825	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) → (𝑧 ∈ 𝐻 ↔ (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) ∈ 𝐻))
40	38, 39	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) → 𝑧 ∈ 𝐻))
41	40	rexlimivv 3180	. . . . . 6 ⊢ (∃𝑥 ∈ ℂ ∃𝑦 ∈ 𝐻 𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) → 𝑧 ∈ 𝐻)
42	34, 41	sylbi 217	. . . . 5 ⊢ (𝑧 ∈ ran ( ·_ℎ ↾ (ℂ × 𝐻)) → 𝑧 ∈ 𝐻)
43	42	ssriv 3926	. . . 4 ⊢ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻
44		df-f 6496	. . . 4 ⊢ (( ·_ℎ ↾ (ℂ × 𝐻)):(ℂ × 𝐻)⟶𝐻 ↔ (( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻) ∧ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻))
45	32, 43, 44	mpbir2an 712	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)):(ℂ × 𝐻)⟶𝐻
46		ax-1cn 11087	. . . . 5 ⊢ 1 ∈ ℂ
47		ovres 7526	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (1 ·_ℎ 𝑥))
48	46, 47	mpan 691	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (1 ·_ℎ 𝑥))
49	1	sheli 31300	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
50		ax-hvmulid 31092	. . . . 5 ⊢ (𝑥 ∈ ℋ → (1 ·_ℎ 𝑥) = 𝑥)
51	49, 50	syl 17	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1 ·_ℎ 𝑥) = 𝑥)
52	48, 51	eqtrd 2772	. . 3 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = 𝑥)
53		id 22	. . . . 5 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
54	1	sheli 31300	. . . . 5 ⊢ (𝑧 ∈ 𝐻 → 𝑧 ∈ ℋ)
55		ax-hvdistr1 31094	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑦 ·_ℎ 𝑧)))
56	53, 49, 54, 55	syl3an 1161	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑦 ·_ℎ 𝑧)))
57		ovres 7526	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧) = (𝑥 +_ℎ 𝑧))
58	57	3adant1 1131	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧) = (𝑥 +_ℎ 𝑧))
59	58	oveq2d 7376	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧)) = (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)))
60		shaddcl 31303	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥 +_ℎ 𝑧) ∈ 𝐻)
61	1, 60	mp3an1 1451	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥 +_ℎ 𝑧) ∈ 𝐻)
62		ovres 7526	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 +_ℎ 𝑧) ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
63	61, 62	sylan2 594	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
64	63	3impb 1115	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
65	59, 64	eqtrd 2772	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
66		ovres 7526	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
67	66	3adant3 1133	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
68		ovres 7526	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧) = (𝑦 ·_ℎ 𝑧))
69	68	3adant2 1132	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧) = (𝑦 ·_ℎ 𝑧))
70	67, 69	oveq12d 7378	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → ((𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧)) = ((𝑦 ·_ℎ 𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑦 ·_ℎ 𝑧)))
71		shmulcl 31304	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
72	1, 71	mp3an1 1451	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
73	72	3adant3 1133	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
74		shmulcl 31304	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
75	1, 74	mp3an1 1451	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
76	75	3adant2 1132	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
77	73, 76	ovresd 7527	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → ((𝑦 ·_ℎ 𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑦 ·_ℎ 𝑧)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑦 ·_ℎ 𝑧)))
78	70, 77	eqtrd 2772	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → ((𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑦 ·_ℎ 𝑧)))
79	56, 65, 78	3eqtr4d 2782	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧)) = ((𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧)))
80		ax-hvdistr2 31095	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·_ℎ 𝑥) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
81	49, 80	syl3an3 1166	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧) ·_ℎ 𝑥) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
82		addcl 11111	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 + 𝑧) ∈ ℂ)
83		ovres 7526	. . . . 5 ⊢ (((𝑦 + 𝑧) ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 + 𝑧) ·_ℎ 𝑥))
84	82, 83	stoic3 1778	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 + 𝑧) ·_ℎ 𝑥))
85	66	3adant2 1132	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
86		ovres 7526	. . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑧 ·_ℎ 𝑥))
87	86	3adant1 1131	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑧 ·_ℎ 𝑥))
88	85, 87	oveq12d 7378	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((𝑦 ·_ℎ 𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑧 ·_ℎ 𝑥)))
89	72	3adant2 1132	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
90		shmulcl 31304	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
91	1, 90	mp3an1 1451	. . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
92	91	3adant1 1131	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
93	89, 92	ovresd 7527	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 ·_ℎ 𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑧 ·_ℎ 𝑥)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
94	88, 93	eqtrd 2772	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
95	81, 84, 94	3eqtr4d 2782	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)( +_ℎ ↾ (𝐻 × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)))
96		ax-hvmulass 31093	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·_ℎ 𝑥) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
97	49, 96	syl3an3 1166	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧) ·_ℎ 𝑥) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
98		mulcl 11113	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · 𝑧) ∈ ℂ)
99		ovres 7526	. . . . 5 ⊢ (((𝑦 · 𝑧) ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 · 𝑧) ·_ℎ 𝑥))
100	98, 99	stoic3 1778	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 · 𝑧) ·_ℎ 𝑥))
101	87	oveq2d 7376	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)))
102		ovres 7526	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑧 ·_ℎ 𝑥) ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
103	91, 102	sylan2 594	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻)) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
104	103	3impb 1115	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
105	101, 104	eqtrd 2772	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
106	97, 100, 105	3eqtr4d 2782	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)))
107		eqid 2737	. . 3 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
108	2, 12, 45, 52, 79, 95, 106, 107	isvciOLD 30666	. 2 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ CVec_OLD
109		normf 31209	. . 3 ⊢ norm_ℎ: ℋ⟶ℝ
110		fssres 6700	. . 3 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ 𝐻 ⊆ ℋ) → (norm_ℎ ↾ 𝐻):𝐻⟶ℝ)
111	109, 5, 110	mp2an 693	. 2 ⊢ (norm_ℎ ↾ 𝐻):𝐻⟶ℝ
112		fvres 6853	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑥) = (norm_ℎ‘𝑥))
113	112	eqeq1d 2739	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (((norm_ℎ ↾ 𝐻)‘𝑥) = 0 ↔ (norm_ℎ‘𝑥) = 0))
114		norm-i 31215	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
115	49, 114	syl 17	. . . 4 ⊢ (𝑥 ∈ 𝐻 → ((norm_ℎ‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
116	113, 115	bitrd 279	. . 3 ⊢ (𝑥 ∈ 𝐻 → (((norm_ℎ ↾ 𝐻)‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
117	116	biimpa 476	. 2 ⊢ ((𝑥 ∈ 𝐻 ∧ ((norm_ℎ ↾ 𝐻)‘𝑥) = 0) → 𝑥 = 0_ℎ)
118		norm-iii 31226	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘(𝑦 ·_ℎ 𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
119	49, 118	sylan2 594	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (norm_ℎ‘(𝑦 ·_ℎ 𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
120	66	fveq2d 6838	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)))
121		fvres 6853	. . . . 5 ⊢ ((𝑦 ·_ℎ 𝑥) ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)) = (norm_ℎ‘(𝑦 ·_ℎ 𝑥)))
122	72, 121	syl 17	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)) = (norm_ℎ‘(𝑦 ·_ℎ 𝑥)))
123	120, 122	eqtrd 2772	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = (norm_ℎ‘(𝑦 ·_ℎ 𝑥)))
124	112	adantl 481	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘𝑥) = (norm_ℎ‘𝑥))
125	124	oveq2d 7376	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
126	119, 123, 125	3eqtr4d 2782	. 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)))
127	1	sheli 31300	. . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ)
128		norm-ii 31224	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (norm_ℎ‘(𝑥 +_ℎ 𝑦)) ≤ ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
129	49, 127, 128	syl2an 597	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (norm_ℎ‘(𝑥 +_ℎ 𝑦)) ≤ ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
130		ovres 7526	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +_ℎ 𝑦))
131	130	fveq2d 6838	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)))
132		shaddcl 31303	. . . . . 6 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
133	1, 132	mp3an1 1451	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
134		fvres 6853	. . . . 5 ⊢ ((𝑥 +_ℎ 𝑦) ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
135	133, 134	syl 17	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
136	131, 135	eqtrd 2772	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
137		fvres 6853	. . . 4 ⊢ (𝑦 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑦) = (norm_ℎ‘𝑦))
138	112, 137	oveqan12d 7379	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (((norm_ℎ ↾ 𝐻)‘𝑥) + ((norm_ℎ ↾ 𝐻)‘𝑦)) = ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
139	129, 136, 138	3brtr4d 5118	. 2 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) ≤ (((norm_ℎ ↾ 𝐻)‘𝑥) + ((norm_ℎ ↾ 𝐻)‘𝑦)))
140		hhssnvt.1	. 2 ⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
141	13, 24, 108, 111, 117, 126, 139, 140	isnvi 30699	1 ⊢ 𝑊 ∈ NrmCVec

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 ⊆ wss 3890 ⟨cop 4574 class class class wbr 5086 × cxp 5622 dom cdm 5624 ran crn 5625 ↾ cres 5626 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 ≤ cle 11171 abscabs 15187 GrpOpcgr 30575 GIdcgi 30576 AbelOpcablo 30630 NrmCVeccnv 30670 ℋchba 31005 +_ℎ cva 31006 ·_ℎ csm 31007 norm_ℎcno 31009 0_ℎc0v 31010 S_ℋ csh 31014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-hilex 31085 ax-hfvadd 31086 ax-hvcom 31087 ax-hvass 31088 ax-hv0cl 31089 ax-hvaddid 31090 ax-hfvmul 31091 ax-hvmulid 31092 ax-hvmulass 31093 ax-hvdistr1 31094 ax-hvdistr2 31095 ax-hvmul0 31096 ax-hfi 31165 ax-his1 31168 ax-his2 31169 ax-his3 31170 ax-his4 31171

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-grpo 30579 df-gid 30580 df-ginv 30581 df-ablo 30631 df-vc 30645 df-nv 30678 df-va 30681 df-ba 30682 df-sm 30683 df-0v 30684 df-nmcv 30686 df-hnorm 31054 df-hba 31055 df-hvsub 31057 df-sh 31293

This theorem is referenced by: hhssnvt 31351 hhssvsf 31359 hhssims 31360 hhssmet 31362 hhssmetdval 31363 hhssbnOLD 31365

W3C validator