riesz3i - Hilbert Space Explorer

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Theorem riesz3i 32133

Description: A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nlelch.1	⊢ 𝑇 ∈ LinFn
nlelch.2	⊢ 𝑇 ∈ ContFn

**Assertion**
Ref	Expression
riesz3i	⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)

Distinct variable group: 𝑤,𝑣,𝑇

Proof of Theorem riesz3i Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hv0cl 31074	. . 3 ⊢ 0_ℎ ∈ ℋ
2		nlelch.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfnfi 32112	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
4		fveq2 6840	. . . . . . . . 9 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (⊥‘(⊥‘(null‘𝑇))) = (⊥‘0_ℋ))
5		nlelch.2	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContFn
6	2, 5	nlelchi 32132	. . . . . . . . . 10 ⊢ (null‘𝑇) ∈ C_ℋ
7	6	ococi 31476	. . . . . . . . 9 ⊢ (⊥‘(⊥‘(null‘𝑇))) = (null‘𝑇)
8		choc0 31397	. . . . . . . . 9 ⊢ (⊥‘0_ℋ) = ℋ
9	4, 7, 8	3eqtr3g 2794	. . . . . . . 8 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (null‘𝑇) = ℋ)
10	9	eleq2d 2822	. . . . . . 7 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (𝑣 ∈ (null‘𝑇) ↔ 𝑣 ∈ ℋ))
11	10	biimpar 477	. . . . . 6 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ (null‘𝑇))
12		elnlfn2 32000	. . . . . 6 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝑣 ∈ (null‘𝑇)) → (𝑇‘𝑣) = 0)
13	3, 11, 12	sylancr 588	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = 0)
14		hi02 31168	. . . . . 6 ⊢ (𝑣 ∈ ℋ → (𝑣 ·_ih 0_ℎ) = 0)
15	14	adantl 481	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 0_ℎ) = 0)
16	13, 15	eqtr4d 2774	. . . 4 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
17	16	ralrimiva 3129	. . 3 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
18		oveq2 7375	. . . . . 6 ⊢ (𝑤 = 0_ℎ → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih 0_ℎ))
19	18	eqeq2d 2747	. . . . 5 ⊢ (𝑤 = 0_ℎ → ((𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
20	19	ralbidv 3160	. . . 4 ⊢ (𝑤 = 0_ℎ → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
21	20	rspcev 3564	. . 3 ⊢ ((0_ℎ ∈ ℋ ∧ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)) → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
22	1, 17, 21	sylancr 588	. 2 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
23	6	choccli 31378	. . . 4 ⊢ (⊥‘(null‘𝑇)) ∈ C_ℋ
24	23	chne0i 31524	. . 3 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ ↔ ∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ)
25	23	cheli 31303	. . . . 5 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → 𝑢 ∈ ℋ)
26	3	ffvelcdmi 7035	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑇‘𝑢) ∈ ℂ)
27	26	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑇‘𝑢) ∈ ℂ)
28		hicl 31151	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
29	28	anidms 566	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑢 ·_ih 𝑢) ∈ ℂ)
30	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
31		his6 31170	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) = 0 ↔ 𝑢 = 0_ℎ))
32	31	necon3bid 2976	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) ≠ 0 ↔ 𝑢 ≠ 0_ℎ))
33	32	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ≠ 0)
34	27, 30, 33	divcld 11931	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
35	34	cjcld 15158	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ)
36		simpl 482	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → 𝑢 ∈ ℋ)
37		hvmulcl 31084	. . . . . . . . 9 ⊢ (((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
38	35, 36, 37	syl2anc 585	. . . . . . . 8 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
39	38	adantll 715	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
40		hvmulcl 31084	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
41	26, 40	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
42	3	ffvelcdmi 7035	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ℋ → (𝑇‘𝑣) ∈ ℂ)
43		hvmulcl 31084	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
44	42, 43	sylan 581	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
45	44	ancoms 458	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
46		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
47		his2sub 31163	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ ∧ ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = ((((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) − (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢)))
48	41, 45, 46, 47	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = ((((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) − (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢)))
49	26	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
50		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
51		ax-his3 31155	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) = ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)))
52	49, 50, 46, 51	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) = ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)))
53	42	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
54		ax-his3 31155	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
55	53, 46, 46, 54	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
56	52, 55	oveq12d 7385	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) − (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢)) = (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))))
57	48, 56	eqtr2d 2772	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢))
58	57	adantll 715	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢))
59		hvsubcl 31088	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ ∧ ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ)
60	41, 45, 59	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ)
61	2	lnfnsubi 32117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ ∧ ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = ((𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) − (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢))))
62	41, 45, 61	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = ((𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) − (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢))))
63	2	lnfnmuli 32115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
64	26, 63	sylan 581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
65	2	lnfnmuli 32115	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑣) · (𝑇‘𝑢)))
66		mulcom 11124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑇‘𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
67	26, 66	sylan2 594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
68	65, 67	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
69	42, 68	sylan 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
70	69	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
71	64, 70	oveq12d 7385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) − (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢))) = (((𝑇‘𝑢) · (𝑇‘𝑣)) − ((𝑇‘𝑢) · (𝑇‘𝑣))))
72		mulcl 11122	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ (𝑇‘𝑣) ∈ ℂ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
73	26, 42, 72	syl2an 597	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
74	73	subidd 11493	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑇‘𝑣)) − ((𝑇‘𝑢) · (𝑇‘𝑣))) = 0)
75	62, 71, 74	3eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0)
76		elnlfn 31999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇: ℋ⟶ℂ → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇) ↔ ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ ∧ (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0)))
77	3, 76	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇) ↔ ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ ∧ (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0))
78	60, 75, 77	sylanbrc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇))
79	6	chssii 31302	. . . . . . . . . . . . . . . . 17 ⊢ (null‘𝑇) ⊆ ℋ
80		ocorth 31362	. . . . . . . . . . . . . . . . 17 ⊢ ((null‘𝑇) ⊆ ℋ → (((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇) ∧ 𝑢 ∈ (⊥‘(null‘𝑇))) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0))
81	79, 80	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇) ∧ 𝑢 ∈ (⊥‘(null‘𝑇))) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
82	78, 81	sylan 581	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) ∧ 𝑢 ∈ (⊥‘(null‘𝑇))) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
83	82	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ (𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ)) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
84	83	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
85	58, 84	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0)
86		hicl 31151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
87	86	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
88	49, 87	mulcld 11165	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
89		mulcl 11122	. . . . . . . . . . . . . . 15 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
90	42, 29, 89	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
91	88, 90	subeq0ad 11515	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0 ↔ ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))))
92	91	adantll 715	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0 ↔ ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))))
93	85, 92	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
94	93	adantlr 716	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
95	88	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
96	42	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
97	30, 33	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
98	97	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
99		divmul3 11814	. . . . . . . . . . . 12 ⊢ ((((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ ∧ (𝑇‘𝑣) ∈ ℂ ∧ ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0)) → ((((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑇‘𝑣) ↔ ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))))
100	95, 96, 98, 99	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑇‘𝑣) ↔ ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))))
101	100	adantlll 719	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑇‘𝑣) ↔ ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))))
102	94, 101	mpbird 257	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑇‘𝑣))
103	27	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
104	87	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
105		div23 11828	. . . . . . . . . . . 12 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ (𝑣 ·_ih 𝑢) ∈ ℂ ∧ ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0)) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) · (𝑣 ·_ih 𝑢)))
106	103, 104, 98, 105	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) · (𝑣 ·_ih 𝑢)))
107	34	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
108		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
109		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
110		his52 31158	. . . . . . . . . . . 12 ⊢ ((((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ ∧ 𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)) = (((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) · (𝑣 ·_ih 𝑢)))
111	107, 108, 109, 110	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)) = (((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) · (𝑣 ·_ih 𝑢)))
112	106, 111	eqtr4d 2774	. . . . . . . . . 10 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
113	112	adantlll 719	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
114	102, 113	eqtr3d 2773	. . . . . . . 8 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
115	114	ralrimiva 3129	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
116		oveq2 7375	. . . . . . . . . 10 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
117	116	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → ((𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢))))
118	117	ralbidv 3160	. . . . . . . 8 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢))))
119	118	rspcev 3564	. . . . . . 7 ⊢ ((((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ ∧ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢))) → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
120	39, 115, 119	syl2anc 585	. . . . . 6 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
121	120	ex 412	. . . . 5 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
122	25, 121	mpdan 688	. . . 4 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
123	122	rexlimiv 3131	. . 3 ⊢ (∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
124	24, 123	sylbi 217	. 2 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
125	22, 124	pm2.61ine 3015	1 ⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 ⊆ wss 3889 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 0cc0 11038 · cmul 11043 − cmin 11377 / cdiv 11807 ∗ccj 15058 ℋchba 30990 ·_ℎ csm 30992 ·_ih csp 30993 0_ℎc0v 30995 −_ℎ cmv 30996 ⊥cort 31001 0_ℋc0h 31006 nullcnl 31023 ContFnccnfn 31024 LinFnclf 31025

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073 ax-hv0cl 31074 ax-hvaddid 31075 ax-hfvmul 31076 ax-hvmulid 31077 ax-hvmulass 31078 ax-hvdistr1 31079 ax-hvdistr2 31080 ax-hvmul0 31081 ax-hfi 31150 ax-his1 31153 ax-his2 31154 ax-his3 31155 ax-his4 31156 ax-hcompl 31273

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-lm 23194 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cfil 25222 df-cau 25223 df-cmet 25224 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672 df-dip 30772 df-ssp 30793 df-ph 30884 df-cbn 30934 df-hnorm 31039 df-hba 31040 df-hvsub 31042 df-hlim 31043 df-hcau 31044 df-sh 31278 df-ch 31292 df-oc 31323 df-ch0 31324 df-nlfn 31917 df-cnfn 31918 df-lnfn 31919

This theorem is referenced by: riesz4i 32134 riesz1 32136

W3C validator