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Theorem hhssnv 31745

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 31743	. . . 4 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp
3		ablogrpo 31028	. . . 4 ⊢ (( +_ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp)
4	2, 3	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp
5	1	shssii 31694	. . . . . 6 ⊢ 𝐻 ⊆ ℋ
6		xpss12 5670	. . . . . 6 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	5, 5, 6	mp2an 705	. . . . 5 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)
8		ax-hfvadd 31481	. . . . . 6 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
9	8	fdmi 6714	. . . . 5 ⊢ dom +_ℎ = ( ℋ × ℋ)
10	7, 9	sseqtrri 3980	. . . 4 ⊢ (𝐻 × 𝐻) ⊆ dom +_ℎ
11		ssdmres 6006	. . . 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 31002	. 2 ⊢ 𝐻 = ran ( +_ℎ ↾ (𝐻 × 𝐻))
14		sh0 31697	. . . . . 6 ⊢ (𝐻 ∈ S_ℋ → 0_ℎ ∈ 𝐻)
15	1, 14	ax-mp 5	. . . . 5 ⊢ 0_ℎ ∈ 𝐻
16		ovres 7579	. . . . 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 31484	. . . . 5 ⊢ 0_ℎ ∈ ℋ
19	18	hvaddlidi 31510	. . . 4 ⊢ (0_ℎ +_ℎ 0_ℎ) = 0_ℎ
20	17, 19	eqtri 2783	. . 3 ⊢ (0_ℎ( +_ℎ ↾ (𝐻 × 𝐻))0_ℎ) = 0_ℎ
21		eqid 2760	. . . . 5 ⊢ (GId‘( +_ℎ ↾ (𝐻 × 𝐻))) = (GId‘( +_ℎ ↾ (𝐻 × 𝐻)))
22	13, 21	grpoid 31001	. . . 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 31486	. . . . . 6 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
26		ffn 6702	. . . . . 6 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
27	25, 26	ax-mp 5	. . . . 5 ⊢ ·_ℎ Fn (ℂ × ℋ)
28		ssid 3953	. . . . . 6 ⊢ ℂ ⊆ ℂ
29		xpss12 5670	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐻 ⊆ ℋ) → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
30	28, 5, 29	mp2an 705	. . . . 5 ⊢ (ℂ × 𝐻) ⊆ (ℂ × ℋ)
31		fnssres 6655	. . . . 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 7590	. . . . . . 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 7579	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) = (𝑥 ·_ℎ 𝑦))
36		shmulcl 31699	. . . . . . . . . 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 2860	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝐻) → (𝑥( ·_ℎ ↾ (ℂ × 𝐻))𝑦) ∈ 𝐻)
39		eleq1 2848	. . . . . . . 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 3204	. . . . . 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 3935	. . . 4 ⊢ ran ( ·_ℎ ↾ (ℂ × 𝐻)) ⊆ 𝐻
44		df-f 6537	. . . 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 11182	. . . . 5 ⊢ 1 ∈ ℂ
47		ovres 7579	. . . . 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 31695	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
50		ax-hvmulid 31487	. . . . 5 ⊢ (𝑥 ∈ ℋ → (1 ·_ℎ 𝑥) = 𝑥)
51	49, 50	syl 18	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (1 ·_ℎ 𝑥) = 𝑥)
52	48, 51	eqtrd 2795	. . 3 ⊢ (𝑥 ∈ 𝐻 → (1( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = 𝑥)
53		id 23	. . . . 5 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
54	1	sheli 31695	. . . . 5 ⊢ (𝑧 ∈ 𝐻 → 𝑧 ∈ ℋ)
55		ax-hvdistr1 31489	. . . . 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 7579	. . . . . . 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 7429	. . . . 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 31698	. . . . . . . 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 7579	. . . . . . 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 2795	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑧)) = (𝑦 ·_ℎ (𝑥 +_ℎ 𝑧)))
66		ovres 7579	. . . . . . 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 7579	. . . . . . 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 7431	. . . . 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 31699	. . . . . . . 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 31699	. . . . . . . 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 7580	. . . . 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 2795	. . . 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 2805	. . 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 31490	. . . . 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 11206	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 + 𝑧) ∈ ℂ)
83		ovres 7579	. . . . 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 7579	. . . . . . 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 7431	. . . . 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 31699	. . . . . . . 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 7580	. . . . 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 2795	. . . 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 2805	. . 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 31488	. . . . 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 11208	. . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · 𝑧) ∈ ℂ)
99		ovres 7579	. . . . 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 7429	. . . . 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 7579	. . . . . . 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 2795	. . . 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 2805	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((𝑦 · 𝑧)( ·_ℎ ↾ (ℂ × 𝐻))𝑥) = (𝑦( ·_ℎ ↾ (ℂ × 𝐻))(𝑧( ·_ℎ ↾ (ℂ × 𝐻))𝑥)))
107		eqid 2760	. . 3 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
108	2, 12, 45, 52, 79, 95, 106, 107	isvciOLD 31061	. 2 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ CVec_OLD
109		normf 31604	. . 3 ⊢ norm_ℎ: ℋ⟶ℝ
110		fssres 6741	. . 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 6897	. . . . 5 ⊢ (𝑥 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑥) = (norm_ℎ‘𝑥))
113	112	eqeq1d 2762	. . . 4 ⊢ (𝑥 ∈ 𝐻 → (((norm_ℎ ↾ 𝐻)‘𝑥) = 0 ↔ (norm_ℎ‘𝑥) = 0))
114		norm-i 31610	. . . . 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 31621	. . . 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 6882	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((norm_ℎ ↾ 𝐻)‘(𝑦 ·_ℎ 𝑥)))
121		fvres 6897	. . . . 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 2795	. . 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 7429	. . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)) = ((abs‘𝑦) · (norm_ℎ‘𝑥)))
126	119, 123, 125	3eqtr4d 2805	. 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑦( ·_ℎ ↾ (ℂ × 𝐻))𝑥)) = ((abs‘𝑦) · ((norm_ℎ ↾ 𝐻)‘𝑥)))
127	1	sheli 31695	. . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ)
128		norm-ii 31619	. . . 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 7579	. . . . 5 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +_ℎ 𝑦))
131	130	fveq2d 6882	. . . 4 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = ((norm_ℎ ↾ 𝐻)‘(𝑥 +_ℎ 𝑦)))
132		shaddcl 31698	. . . . . 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 6897	. . . . 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 2795	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → ((norm_ℎ ↾ 𝐻)‘(𝑥( +_ℎ ↾ (𝐻 × 𝐻))𝑦)) = (norm_ℎ‘(𝑥 +_ℎ 𝑦)))
137		fvres 6897	. . . 4 ⊢ (𝑦 ∈ 𝐻 → ((norm_ℎ ↾ 𝐻)‘𝑦) = (norm_ℎ‘𝑦))
138	112, 137	oveqan12d 7432	. . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (((norm_ℎ ↾ 𝐻)‘𝑥) + ((norm_ℎ ↾ 𝐻)‘𝑦)) = ((norm_ℎ‘𝑥) + (norm_ℎ‘𝑦)))
139	129, 136, 138	3brtr4d 5137	. 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 31094	1 ⊢ 𝑊 ∈ NrmCVec

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3086 ⊆ wss 3899 ⟨cop 4590 class class class wbr 5103 × cxp 5653 dom cdm 5655 ran crn 5656 ↾ cres 5657 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 ≤ cle 11268 abscabs 15321 GrpOpcgr 30970 GIdcgi 30971 AbelOpcablo 31025 NrmCVeccnv 31065 ℋchba 31400 +_ℎ cva 31401 ·_ℎ csm 31402 norm_ℎcno 31404 0_ℎc0v 31405 S_ℋ csh 31409

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-hilex 31480 ax-hfvadd 31481 ax-hvcom 31482 ax-hvass 31483 ax-hv0cl 31484 ax-hvaddid 31485 ax-hfvmul 31486 ax-hvmulid 31487 ax-hvmulass 31488 ax-hvdistr1 31489 ax-hvdistr2 31490 ax-hvmul0 31491 ax-hfi 31560 ax-his1 31563 ax-his2 31564 ax-his3 31565 ax-his4 31566

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-n0 12529 df-z 12616 df-uz 12888 df-rp 13043 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-grpo 30974 df-gid 30975 df-ginv 30976 df-ablo 31026 df-vc 31040 df-nv 31073 df-va 31076 df-ba 31077 df-sm 31078 df-0v 31079 df-nmcv 31081 df-hnorm 31449 df-hba 31450 df-hvsub 31452 df-sh 31688

This theorem is used by: hhssnvt 31746 hhssvsf 31754 hhssims 31755 hhssmet 31757 hhssmetdval 31758 hhssbnOLD 31760

W3C validator