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Theorem hhssnv 31687

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 31685	. . . 4 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp
3		ablogrpo 30970	. . . 4 ⊢ (( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp)
4	2, 3	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp
5	1	shssii 31636	. . . . . 6 ⊢ 𝐻 ⊆ ℋ
6		xpss12 5678	. . . . . 6 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	5, 5, 6	mp2an 705	. . . . 5 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)
8		ax-hfvadd 31423	. . . . . 6 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
9	8	fdmi 6721	. . . . 5 ⊢ dom +_ℎ = ( ℋ × ℋ)
10	7, 9	sseqtrri 3987	. . . 4 ⊢ (𝐻 × 𝐻) ⊆ dom +_ℎ
11		ssdmres 6014	. . . 4 ⊢ ((𝐻 × 𝐻) ⊆ dom +_ℎ ↔ dom ( +_ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻))
12	10, 11	mpbi 233	. . 3 ⊢ dom ( +_ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻)
13	4, 12	grporn 30944	. 2 ⊢ 𝐻 = ran ( +_ℎ ↾ (𝐻 × 𝐻))
14		sh0 31639	. . . . . 6 ⊢ (𝐻 ∈ S_ℋ → 0_ℎ ∈ 𝐻)
15	1, 14	ax-mp 5	. . . . 5 ⊢ 0_ℎ ∈ 𝐻
16		ovres 7585	. . . . 5 ⊢ ((0_ℎ ∈ 𝐻 ∧ 0_ℎ ∈ 𝐻) → (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = (0_ℎ +_ℎ 0_ℎ))
17	15, 15, 16	mp2an 705	. . . 4 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = (0_ℎ +_ℎ 0_ℎ)
18		ax-hv0cl 31426	. . . . 5 ⊢ 0_ℎ ∈ ℋ
19	18	hvaddlidi 31452	. . . 4 ⊢ (0_ℎ +_ℎ 0_ℎ) = 0_ℎ
20	17, 19	eqtri 2788	. . 3 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ
21		eqid 2765	. . . . 5 ⊢ (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) = (GId‘( +_ℎ ↾ (𝐻 × 𝐻)))
22	13, 21	grpoid 30943	. . . 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 705	. . 3 ⊢ (0_ℎ = (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) ↔ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ)
24	20, 23	mpbir 234	. 2 ⊢ 0_ℎ = (GId‘( +_ℎ ↾ (𝐻 × 𝐻)))
25		ax-hfvmul 31428	. . . . . 6 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
26		ffn 6709	. . . . . 6 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
27	25, 26	ax-mp 5	. . . . 5 ⊢ ·_ℎ Fn (ℂ × ℋ)
28		ssid 3960	. . . . . 6 ⊢ ℂ ⊆ ℂ
29		xpss12 5678	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐻 ⊆ ℋ) → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
30	28, 5, 29	mp2an 705	. . . . 5 ⊢ (ℂ × 𝐻) ⊆ (ℂ × ℋ)
31		fnssres 6662	. . . . 5 ⊢ (( ·_ℎ Fn (ℂ × ℋ) ∧ (ℂ × 𝐻) ⊆ (ℂ × ℋ)) → ( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻))
32	27, 30, 31	mp2an 705	. . . 4 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻)
33		ovelrn 7596	. . . . . . 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 7585	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) = (𝑥 ·_ℎ 𝑦))
36		shmulcl 31641	. . . . . . . . . 10 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
37	1, 36	mp3an1 1477	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
38	35, 37	eqeltrd 2865	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) ∈ 𝐻)
39		eleq1 2853	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) → (𝑧 ∈ 𝐻 ↔ (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) ∈ 𝐻))
40	38, 39	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) → 𝑧 ∈ 𝐻))
41	40	rexlimivv 3209	. . . . . 6 ⊢ (∃𝑥 ∈ ℂ ∃𝑦 ∈ 𝐻 𝑧 = (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) → 𝑧 ∈ 𝐻)
42	34, 41	sylbi 220	. . . . 5 ⊢ (𝑧 ∈ ran ( ·_ℎ ↾ (ℂ × 𝐻)) → 𝑧 ∈ 𝐻)
43	42	ssriv 3942	. . . 4 ⊢ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻
44		df-f 6544	. . . 4 ⊢ (( ·_ℎ ↾ (ℂ × 𝐻)):(ℂ × 𝐻)⟶𝐻 ↔ (( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻) ∧ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻))
45	32, 43, 44	mpbir2an 724	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)):(ℂ × 𝐻)⟶𝐻
46		ax-1cn 11173	. . . . 5 ⊢ 1 ∈ ℂ
47		ovres 7585	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (1 ·_ℎ 𝑥))
48	46, 47	mpan 703	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (1 ·_ℎ 𝑥))
49	1	sheli 31637	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
50		ax-hvmulid 31429	. . . . 5 ⊢ (𝑥 ∈ ℋ → (1 ·_ℎ 𝑥) = 𝑥)
51	49, 50	syl 18	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1 ·_ℎ 𝑥) = 𝑥)
52	48, 51	eqtrd 2800	. . 3 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = 𝑥)
53		id 23	. . . . 5 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
54	1	sheli 31637	. . . . 5 ⊢ (𝑧 ∈ 𝐻 → 𝑧 ∈ ℋ)
55		ax-hvdistr1 31431	. . . . 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 1178	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑦 ·_ℎ 𝑧)))
57		ovres 7585	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧) = (𝑥 +_ℎ 𝑧))
58	57	3adant1 1148	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧) = (𝑥 +_ℎ 𝑧))
59	58	oveq2d 7435	. . . . 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 31640	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥 +_ℎ 𝑧) ∈ 𝐻)
61	1, 60	mp3an1 1477	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥 +_ℎ 𝑧) ∈ 𝐻)
62		ovres 7585	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 +_ℎ 𝑧) ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
63	61, 62	sylan2 605	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
64	63	3impb 1132	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
65	59, 64	eqtrd 2800	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
66		ovres 7585	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
67	66	3adant3 1150	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
68		ovres 7585	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧) = (𝑦 ·_ℎ 𝑧))
69	68	3adant2 1149	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧) = (𝑦 ·_ℎ 𝑧))
70	67, 69	oveq12d 7437	. . . . 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 31641	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
72	1, 71	mp3an1 1477	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
73	72	3adant3 1150	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
74		shmulcl 31641	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
75	1, 74	mp3an1 1477	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
76	75	3adant2 1149	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
77	73, 76	ovresd 7586	. . . . 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 2800	. . . 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 2810	. . 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 31432	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·_ℎ 𝑥) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
81	49, 80	syl3an3 1183	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧) ·_ℎ 𝑥) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
82		addcl 11197	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 + 𝑧) ∈ ℂ)
83		ovres 7585	. . . . 5 ⊢ (((𝑦 + 𝑧) ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 + 𝑧) ·_ℎ 𝑥))
84	82, 83	stoic3 1809	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 + 𝑧) ·_ℎ 𝑥))
85	66	3adant2 1149	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
86		ovres 7585	. . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑧 ·_ℎ 𝑥))
87	86	3adant1 1148	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑧 ·_ℎ 𝑥))
88	85, 87	oveq12d 7437	. . . . 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 1149	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
90		shmulcl 31641	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
91	1, 90	mp3an1 1477	. . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
92	91	3adant1 1148	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
93	89, 92	ovresd 7586	. . . . 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 2800	. . . 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 2810	. . 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 31430	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·_ℎ 𝑥) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
97	49, 96	syl3an3 1183	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧) ·_ℎ 𝑥) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
98		mulcl 11199	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · 𝑧) ∈ ℂ)
99		ovres 7585	. . . . 5 ⊢ (((𝑦 · 𝑧) ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 · 𝑧) ·_ℎ 𝑥))
100	98, 99	stoic3 1809	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 · 𝑧) ·_ℎ 𝑥))
101	87	oveq2d 7435	. . . . 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 7585	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑧 ·_ℎ 𝑥) ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
103	91, 102	sylan2 605	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻)) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
104	103	3impb 1132	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
105	101, 104	eqtrd 2800	. . . 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 2810	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)))
107		eqid 2765	. . 3 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
108	2, 12, 45, 52, 79, 95, 106, 107	isvciOLD 31003	. 2 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ CVec_OLD
109		normf 31546	. . 3 ⊢ norm_ℎ: ℋ⟶ℝ
110		fssres 6748	. . 3 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ 𝐻 ⊆ ℋ) → (norm_ℎ ↾ 𝐻):𝐻⟶ℝ)
111	109, 5, 110	mp2an 705	. 2 ⊢ (norm_ℎ ↾ 𝐻):𝐻⟶ℝ
112		fvres 6904	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑥) = (norm_ℎ‘𝑥))
113	112	eqeq1d 2767	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (((norm_ℎ ↾ 𝐻)‘𝑥) = 0 ↔ (norm_ℎ‘𝑥) = 0))
114		norm-i 31552	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
115	49, 114	syl 18	. . . 4 ⊢ (𝑥 ∈ 𝐻 → ((norm_ℎ‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
116	113, 115	bitrd 282	. . 3 ⊢ (𝑥 ∈ 𝐻 → (((norm_ℎ ↾ 𝐻)‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
117	116	biimpa 482	. 2 ⊢ ((𝑥 ∈ 𝐻 ∧ ((norm_ℎ ↾ 𝐻)‘𝑥) = 0) → 𝑥 = 0_ℎ)
118		norm-iii 31563	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘(𝑦 ·_ℎ 𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
119	49, 118	sylan2 605	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (norm_ℎ‘(𝑦 ·_ℎ 𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
120	66	fveq2d 6889	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)))
121		fvres 6904	. . . . 5 ⊢ ((𝑦 ·_ℎ 𝑥) ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)) = (norm_ℎ‘(𝑦 ·_ℎ 𝑥)))
122	72, 121	syl 18	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)) = (norm_ℎ‘(𝑦 ·_ℎ 𝑥)))
123	120, 122	eqtrd 2800	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = (norm_ℎ‘(𝑦 ·_ℎ 𝑥)))
124	112	adantl 487	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘𝑥) = (norm_ℎ‘𝑥))
125	124	oveq2d 7435	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
126	119, 123, 125	3eqtr4d 2810	. 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)))
127	1	sheli 31637	. . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ)
128		norm-ii 31561	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (norm_ℎ‘(𝑥 +_ℎ 𝑦)) ≤ ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
129	49, 127, 128	syl2an 608	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (norm_ℎ‘(𝑥 +_ℎ 𝑦)) ≤ ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
130		ovres 7585	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +_ℎ 𝑦))
131	130	fveq2d 6889	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)))
132		shaddcl 31640	. . . . . 6 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
133	1, 132	mp3an1 1477	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
134		fvres 6904	. . . . 5 ⊢ ((𝑥 +_ℎ 𝑦) ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
135	133, 134	syl 18	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
136	131, 135	eqtrd 2800	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
137		fvres 6904	. . . 4 ⊢ (𝑦 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑦) = (norm_ℎ‘𝑦))
138	112, 137	oveqan12d 7438	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (((norm_ℎ ↾ 𝐻)‘𝑥) + ((norm_ℎ ↾ 𝐻)‘𝑦)) = ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
139	129, 136, 138	3brtr4d 5145	. 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 31036	1 ⊢ 𝑊 ∈ NrmCVec

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 ⊆ wss 3906 ⟨cop 4597 class class class wbr 5111 × cxp 5661 dom cdm 5663 ran crn 5664 ↾ cres 5665 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 ≤ cle 11259 abscabs 15309 GrpOpcgr 30912 GIdcgi 30913 AbelOpcablo 30967 NrmCVeccnv 31007 ℋchba 31342 +_ℎ cva 31343 ·_ℎ csm 31344 norm_ℎcno 31346 0_ℎc0v 31347 S_ℋ csh 31351

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-hilex 31422 ax-hfvadd 31423 ax-hvcom 31424 ax-hvass 31425 ax-hv0cl 31426 ax-hvaddid 31427 ax-hfvmul 31428 ax-hvmulid 31429 ax-hvmulass 31430 ax-hvdistr1 31431 ax-hvdistr2 31432 ax-hvmul0 31433 ax-hfi 31502 ax-his1 31505 ax-his2 31506 ax-his3 31507 ax-his4 31508

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-sup 9409 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-n0 12520 df-z 12607 df-uz 12879 df-rp 13033 df-seq 14056 df-exp 14116 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-grpo 30916 df-gid 30917 df-ginv 30918 df-ablo 30968 df-vc 30982 df-nv 31015 df-va 31018 df-ba 31019 df-sm 31020 df-0v 31021 df-nmcv 31023 df-hnorm 31391 df-hba 31392 df-hvsub 31394 df-sh 31630

This theorem is used by: hhssnvt 31688 hhssvsf 31696 hhssims 31697 hhssmet 31699 hhssmetdval 31700 hhssbnOLD 31702

W3C validator