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Theorem hhssnv 31292

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 31290	. . . 4 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp
3		ablogrpo 30575	. . . 4 ⊢ (( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp)
4	2, 3	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp
5	1	shssii 31241	. . . . . 6 ⊢ 𝐻 ⊆ ℋ
6		xpss12 5703	. . . . . 6 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	5, 5, 6	mp2an 692	. . . . 5 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)
8		ax-hfvadd 31028	. . . . . 6 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
9	8	fdmi 6747	. . . . 5 ⊢ dom +_ℎ = ( ℋ × ℋ)
10	7, 9	sseqtrri 4032	. . . 4 ⊢ (𝐻 × 𝐻) ⊆ dom +_ℎ
11		ssdmres 6032	. . . 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 30549	. 2 ⊢ 𝐻 = ran ( +_ℎ ↾ (𝐻 × 𝐻))
14		sh0 31244	. . . . . 6 ⊢ (𝐻 ∈ S_ℋ → 0_ℎ ∈ 𝐻)
15	1, 14	ax-mp 5	. . . . 5 ⊢ 0_ℎ ∈ 𝐻
16		ovres 7598	. . . . 5 ⊢ ((0_ℎ ∈ 𝐻 ∧ 0_ℎ ∈ 𝐻) → (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = (0_ℎ +_ℎ 0_ℎ))
17	15, 15, 16	mp2an 692	. . . 4 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = (0_ℎ +_ℎ 0_ℎ)
18		ax-hv0cl 31031	. . . . 5 ⊢ 0_ℎ ∈ ℋ
19	18	hvaddlidi 31057	. . . 4 ⊢ (0_ℎ +_ℎ 0_ℎ) = 0_ℎ
20	17, 19	eqtri 2762	. . 3 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ
21		eqid 2734	. . . . 5 ⊢ (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) = (GId‘( +_ℎ ↾ (𝐻 × 𝐻)))
22	13, 21	grpoid 30548	. . . 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 692	. . 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 31033	. . . . . 6 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
26		ffn 6736	. . . . . 6 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
27	25, 26	ax-mp 5	. . . . 5 ⊢ ·_ℎ Fn (ℂ × ℋ)
28		ssid 4017	. . . . . 6 ⊢ ℂ ⊆ ℂ
29		xpss12 5703	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐻 ⊆ ℋ) → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
30	28, 5, 29	mp2an 692	. . . . 5 ⊢ (ℂ × 𝐻) ⊆ (ℂ × ℋ)
31		fnssres 6691	. . . . 5 ⊢ (( ·_ℎ Fn (ℂ × ℋ) ∧ (ℂ × 𝐻) ⊆ (ℂ × ℋ)) → ( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻))
32	27, 30, 31	mp2an 692	. . . 4 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻)
33		ovelrn 7608	. . . . . . 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 7598	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) = (𝑥 ·_ℎ 𝑦))
36		shmulcl 31246	. . . . . . . . . 10 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
37	1, 36	mp3an1 1447	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥 ·_ℎ 𝑦) ∈ 𝐻)
38	35, 37	eqeltrd 2838	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) ∈ 𝐻)
39		eleq1 2826	. . . . . . . 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 3198	. . . . . 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 3998	. . . 4 ⊢ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻
44		df-f 6566	. . . 4 ⊢ (( ·_ℎ ↾ (ℂ × 𝐻)):(ℂ × 𝐻)⟶𝐻 ↔ (( ·_ℎ ↾ (ℂ × 𝐻)) Fn (ℂ × 𝐻) ∧ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻))
45	32, 43, 44	mpbir2an 711	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)):(ℂ × 𝐻)⟶𝐻
46		ax-1cn 11210	. . . . 5 ⊢ 1 ∈ ℂ
47		ovres 7598	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (1 ·_ℎ 𝑥))
48	46, 47	mpan 690	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (1 ·_ℎ 𝑥))
49	1	sheli 31242	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
50		ax-hvmulid 31034	. . . . 5 ⊢ (𝑥 ∈ ℋ → (1 ·_ℎ 𝑥) = 𝑥)
51	49, 50	syl 17	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1 ·_ℎ 𝑥) = 𝑥)
52	48, 51	eqtrd 2774	. . 3 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = 𝑥)
53		id 22	. . . . 5 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
54	1	sheli 31242	. . . . 5 ⊢ (𝑧 ∈ 𝐻 → 𝑧 ∈ ℋ)
55		ax-hvdistr1 31036	. . . . 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 1159	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑦 ·_ℎ 𝑧)))
57		ovres 7598	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧) = (𝑥 +_ℎ 𝑧))
58	57	3adant1 1129	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧) = (𝑥 +_ℎ 𝑧))
59	58	oveq2d 7446	. . . . 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 31245	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥 +_ℎ 𝑧) ∈ 𝐻)
61	1, 60	mp3an1 1447	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑥 +_ℎ 𝑧) ∈ 𝐻)
62		ovres 7598	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 +_ℎ 𝑧) ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
63	61, 62	sylan2 593	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
64	63	3impb 1114	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥 +_ℎ 𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
65	59, 64	eqtrd 2774	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
66		ovres 7598	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
67	66	3adant3 1131	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
68		ovres 7598	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧) = (𝑦 ·_ℎ 𝑧))
69	68	3adant2 1130	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑧) = (𝑦 ·_ℎ 𝑧))
70	67, 69	oveq12d 7448	. . . . 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 31246	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
72	1, 71	mp3an1 1447	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
73	72	3adant3 1131	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
74		shmulcl 31246	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
75	1, 74	mp3an1 1447	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
76	75	3adant2 1130	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦 ·_ℎ 𝑧) ∈ 𝐻)
77	73, 76	ovresd 7599	. . . . 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 2774	. . . 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 2784	. . 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 31037	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 + 𝑧) ·_ℎ 𝑥) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
81	49, 80	syl3an3 1164	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧) ·_ℎ 𝑥) = ((𝑦 ·_ℎ 𝑥) +_ℎ (𝑧 ·_ℎ 𝑥)))
82		addcl 11234	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 + 𝑧) ∈ ℂ)
83		ovres 7598	. . . . 5 ⊢ (((𝑦 + 𝑧) ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 + 𝑧) ·_ℎ 𝑥))
84	82, 83	stoic3 1772	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 + 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 + 𝑧) ·_ℎ 𝑥))
85	66	3adant2 1130	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦 ·_ℎ 𝑥))
86		ovres 7598	. . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑧 ·_ℎ 𝑥))
87	86	3adant1 1129	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑧 ·_ℎ 𝑥))
88	85, 87	oveq12d 7448	. . . . 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 1130	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦 ·_ℎ 𝑥) ∈ 𝐻)
90		shmulcl 31246	. . . . . . . 8 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
91	1, 90	mp3an1 1447	. . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
92	91	3adant1 1129	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑧 ·_ℎ 𝑥) ∈ 𝐻)
93	89, 92	ovresd 7599	. . . . 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 2774	. . . 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 2784	. . 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 31035	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 · 𝑧) ·_ℎ 𝑥) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
97	49, 96	syl3an3 1164	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧) ·_ℎ 𝑥) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
98		mulcl 11236	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · 𝑧) ∈ ℂ)
99		ovres 7598	. . . . 5 ⊢ (((𝑦 · 𝑧) ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 · 𝑧) ·_ℎ 𝑥))
100	98, 99	stoic3 1772	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = ((𝑦 · 𝑧) ·_ℎ 𝑥))
101	87	oveq2d 7446	. . . . 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 7598	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑧 ·_ℎ 𝑥) ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
103	91, 102	sylan2 593	. . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻)) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
104	103	3impb 1114	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧 ·_ℎ 𝑥)) = (𝑦 ·_ℎ (𝑧 ·_ℎ 𝑥)))
105	101, 104	eqtrd 2774	. . . 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 2784	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)))
107		eqid 2734	. . 3 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
108	2, 12, 45, 52, 79, 95, 106, 107	isvciOLD 30608	. 2 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ CVec_OLD
109		normf 31151	. . 3 ⊢ norm_ℎ: ℋ⟶ℝ
110		fssres 6774	. . 3 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ 𝐻 ⊆ ℋ) → (norm_ℎ ↾ 𝐻):𝐻⟶ℝ)
111	109, 5, 110	mp2an 692	. 2 ⊢ (norm_ℎ ↾ 𝐻):𝐻⟶ℝ
112		fvres 6925	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑥) = (norm_ℎ‘𝑥))
113	112	eqeq1d 2736	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (((norm_ℎ ↾ 𝐻)‘𝑥) = 0 ↔ (norm_ℎ‘𝑥) = 0))
114		norm-i 31157	. . . . 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 31168	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘(𝑦 ·_ℎ 𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
119	49, 118	sylan2 593	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → (norm_ℎ‘(𝑦 ·_ℎ 𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
120	66	fveq2d 6910	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)))
121		fvres 6925	. . . . 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 2774	. . 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 7446	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
126	119, 123, 125	3eqtr4d 2784	. 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)))
127	1	sheli 31242	. . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ)
128		norm-ii 31166	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (norm_ℎ‘(𝑥 +_ℎ 𝑦)) ≤ ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
129	49, 127, 128	syl2an 596	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (norm_ℎ‘(𝑥 +_ℎ 𝑦)) ≤ ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
130		ovres 7598	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +_ℎ 𝑦))
131	130	fveq2d 6910	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)))
132		shaddcl 31245	. . . . . 6 ⊢ ((𝐻 ∈ S_ℋ ∧ 𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
133	1, 132	mp3an1 1447	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +_ℎ 𝑦) ∈ 𝐻)
134		fvres 6925	. . . . 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 2774	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
137		fvres 6925	. . . 4 ⊢ (𝑦 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑦) = (norm_ℎ‘𝑦))
138	112, 137	oveqan12d 7449	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (((norm_ℎ ↾ 𝐻)‘𝑥) + ((norm_ℎ ↾ 𝐻)‘𝑦)) = ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
139	129, 136, 138	3brtr4d 5179	. 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 30641	1 ⊢ 𝑊 ∈ NrmCVec

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ⊆ wss 3962 ⟨cop 4636 class class class wbr 5147 × cxp 5686 dom cdm 5688 ran crn 5689 ↾ cres 5690 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ≤ cle 11293 abscabs 15269 GrpOpcgr 30517 GIdcgi 30518 AbelOpcablo 30572 NrmCVeccnv 30612 ℋchba 30947 +_ℎ cva 30948 ·_ℎ csm 30949 norm_ℎcno 30951 0_ℎc0v 30952 S_ℋ csh 30956

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-hilex 31027 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvmulass 31035 ax-hvdistr1 31036 ax-hvdistr2 31037 ax-hvmul0 31038 ax-hfi 31107 ax-his1 31110 ax-his2 31111 ax-his3 31112 ax-his4 31113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-grpo 30521 df-gid 30522 df-ginv 30523 df-ablo 30573 df-vc 30587 df-nv 30620 df-va 30623 df-ba 30624 df-sm 30625 df-0v 30626 df-nmcv 30628 df-hnorm 30996 df-hba 30997 df-hvsub 30999 df-sh 31235

This theorem is referenced by: hhssnvt 31293 hhssvsf 31301 hhssims 31302 hhssmet 31304 hhssmetdval 31305 hhssbnOLD 31307

W3C validator