riesz3i - Hilbert Space Explorer

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Theorem riesz3i 32151

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 31092	. . 3 ⊢ 0_ℎ ∈ ℋ
2		nlelch.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfnfi 32130	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
4		fveq2 6827	. . . . . . . . 9 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (⊥‘(⊥‘(null‘𝑇))) = (⊥‘0_ℋ))
5		nlelch.2	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContFn
6	2, 5	nlelchi 32150	. . . . . . . . . 10 ⊢ (null‘𝑇) ∈ C_ℋ
7	6	ococi 31494	. . . . . . . . 9 ⊢ (⊥‘(⊥‘(null‘𝑇))) = (null‘𝑇)
8		choc0 31415	. . . . . . . . 9 ⊢ (⊥‘0_ℋ) = ℋ
9	4, 7, 8	3eqtr3g 2797	. . . . . . . 8 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (null‘𝑇) = ℋ)
10	9	eleq2d 2825	. . . . . . 7 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (𝑣 ∈ (null‘𝑇) ↔ 𝑣 ∈ ℋ))
11	10	biimpar 478	. . . . . 6 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ (null‘𝑇))
12		elnlfn2 32018	. . . . . 6 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝑣 ∈ (null‘𝑇)) → (𝑇‘𝑣) = 0)
13	3, 11, 12	sylancr 593	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = 0)
14		hi02 31186	. . . . . 6 ⊢ (𝑣 ∈ ℋ → (𝑣 ·_ih 0_ℎ) = 0)
15	14	adantl 482	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 0_ℎ) = 0)
16	13, 15	eqtr4d 2777	. . . 4 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
17	16	ralrimiva 3131	. . 3 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
18		oveq2 7364	. . . . . 6 ⊢ (𝑤 = 0_ℎ → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih 0_ℎ))
19	18	eqeq2d 2750	. . . . 5 ⊢ (𝑤 = 0_ℎ → ((𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
20	19	ralbidv 3162	. . . 4 ⊢ (𝑤 = 0_ℎ → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
21	20	rspcev 3560	. . 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 593	. 2 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
23	6	choccli 31396	. . . 4 ⊢ (⊥‘(null‘𝑇)) ∈ C_ℋ
24	23	chne0i 31542	. . 3 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ ↔ ∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ)
25	23	cheli 31321	. . . . 5 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → 𝑢 ∈ ℋ)
26	3	ffvelcdmi 7024	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑇‘𝑢) ∈ ℂ)
27	26	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑇‘𝑢) ∈ ℂ)
28		hicl 31169	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
29	28	anidms 571	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑢 ·_ih 𝑢) ∈ ℂ)
30	29	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
31		his6 31188	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) = 0 ↔ 𝑢 = 0_ℎ))
32	31	necon3bid 2978	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) ≠ 0 ↔ 𝑢 ≠ 0_ℎ))
33	32	biimpar 478	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ≠ 0)
34	27, 30, 33	divcld 11922	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
35	34	cjcld 15149	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ)
36		simpl 483	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → 𝑢 ∈ ℋ)
37		hvmulcl 31102	. . . . . . . . 9 ⊢ (((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
38	35, 36, 37	syl2anc 590	. . . . . . . 8 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
39	38	adantll 720	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
40		hvmulcl 31102	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
41	26, 40	sylan 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
42	3	ffvelcdmi 7024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ℋ → (𝑇‘𝑣) ∈ ℂ)
43		hvmulcl 31102	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
44	42, 43	sylan 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
45	44	ancoms 459	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
46		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
47		his2sub 31181	. . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
50		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
51		ax-his3 31173	. . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) = ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)))
53	42	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
54		ax-his3 31173	. . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
56	52, 55	oveq12d 7374	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 720	. . . . . . . . . . . . 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 31106	. . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ)
61	2	lnfnsubi 32135	. . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . 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 32133	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
64	26, 63	sylan 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
65	2	lnfnmuli 32133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑣) · (𝑇‘𝑢)))
66		mulcom 11115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑇‘𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
67	26, 66	sylan2 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
68	65, 67	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
69	42, 68	sylan 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
70	69	ancoms 459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
71	64, 70	oveq12d 7374	. . . . . . . . . . . . . . . . . 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 11113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ (𝑇‘𝑣) ∈ ℂ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
73	26, 42, 72	syl2an 602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
74	73	subidd 11484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑇‘𝑣)) − ((𝑇‘𝑢) · (𝑇‘𝑣))) = 0)
75	62, 71, 74	3eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0)
76		elnlfn 32017	. . . . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇))
79	6	chssii 31320	. . . . . . . . . . . . . . . . 17 ⊢ (null‘𝑇) ⊆ ℋ
80		ocorth 31380	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) ∧ 𝑢 ∈ (⊥‘(null‘𝑇))) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
83	82	ancoms 459	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ (𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ)) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
84	83	anassrs 468	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
85	58, 84	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0)
86		hicl 31169	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
87	86	ancoms 459	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
88	49, 87	mulcld 11156	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
89		mulcl 11113	. . . . . . . . . . . . . . 15 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
90	42, 29, 89	syl2anr 603	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
91	88, 90	subeq0ad 11506	. . . . . . . . . . . . 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 720	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
94	93	adantlr 721	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
95	88	adantlr 721	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
96	42	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
97	30, 33	jca 516	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
98	97	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
99		divmul3 11805	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 724	. . . . . . . . . 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 258	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑇‘𝑣))
103	27	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
104	87	adantlr 721	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
105		div23 11819	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
108		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
109		simpll 772	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
110		his52 31176	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 2777	. . . . . . . . . 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 724	. . . . . . . . 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 2776	. . . . . . . 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 3131	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
116		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
117	116	eqeq2d 2750	. . . . . . . . 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 3162	. . . . . . . 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 3560	. . . . . . 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 590	. . . . . 6 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
121	120	ex 413	. . . . 5 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
122	25, 121	mpdan 693	. . . 4 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
123	122	rexlimiv 3133	. . 3 ⊢ (∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
124	24, 123	sylbi 218	. 2 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
125	22, 124	pm2.61ine 3017	1 ⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 ⊆ wss 3883 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 0cc0 11029 · cmul 11034 − cmin 11368 / cdiv 11798 ∗ccj 15049 ℋchba 31008 ·_ℎ csm 31010 ·_ih csp 31011 0_ℎc0v 31013 −_ℎ cmv 31014 ⊥cort 31019 0_ℋc0h 31024 nullcnl 31041 ContFnccnfn 31042 LinFnclf 31043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 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-addf 11108 ax-mulf 11109 ax-hilex 31088 ax-hfvadd 31089 ax-hvcom 31090 ax-hvass 31091 ax-hv0cl 31092 ax-hvaddid 31093 ax-hfvmul 31094 ax-hvmulid 31095 ax-hvmulass 31096 ax-hvdistr1 31097 ax-hvdistr2 31098 ax-hvmul0 31099 ax-hfi 31168 ax-his1 31171 ax-his2 31172 ax-his3 31173 ax-his4 31174 ax-hcompl 31291

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-acn 9857 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-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-cn 23210 df-cnp 23211 df-lm 23212 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cfil 25240 df-cau 25241 df-cmet 25242 df-grpo 30582 df-gid 30583 df-ginv 30584 df-gdiv 30585 df-ablo 30634 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-vs 30688 df-nmcv 30689 df-ims 30690 df-dip 30790 df-ssp 30811 df-ph 30902 df-cbn 30952 df-hnorm 31057 df-hba 31058 df-hvsub 31060 df-hlim 31061 df-hcau 31062 df-sh 31296 df-ch 31310 df-oc 31341 df-ch0 31342 df-nlfn 31935 df-cnfn 31936 df-lnfn 31937

This theorem is referenced by: riesz4i 32152 riesz1 32154

W3C validator