riesz3i - Hilbert Space Explorer

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Theorem riesz3i 32355

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 31296	. . 3 ⊢ 0_ℎ ∈ ℋ
2		nlelch.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfnfi 32334	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
4		fveq2 6882	. . . . . . . . 9 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (⊥‘(⊥‘(null‘𝑇))) = (⊥‘0_ℋ))
5		nlelch.2	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContFn
6	2, 5	nlelchi 32354	. . . . . . . . . 10 ⊢ (null‘𝑇) ∈ C_ℋ
7	6	ococi 31698	. . . . . . . . 9 ⊢ (⊥‘(⊥‘(null‘𝑇))) = (null‘𝑇)
8		choc0 31619	. . . . . . . . 9 ⊢ (⊥‘0_ℋ) = ℋ
9	4, 7, 8	3eqtr3g 2827	. . . . . . . 8 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (null‘𝑇) = ℋ)
10	9	eleq2d 2855	. . . . . . 7 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (𝑣 ∈ (null‘𝑇) ↔ 𝑣 ∈ ℋ))
11	10	biimpar 482	. . . . . 6 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ (null‘𝑇))
12		elnlfn2 32222	. . . . . 6 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝑣 ∈ (null‘𝑇)) → (𝑇‘𝑣) = 0)
13	3, 11, 12	sylancr 598	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = 0)
14		hi02 31390	. . . . . 6 ⊢ (𝑣 ∈ ℋ → (𝑣 ·_ih 0_ℎ) = 0)
15	14	adantl 486	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 0_ℎ) = 0)
16	13, 15	eqtr4d 2807	. . . 4 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
17	16	ralrimiva 3163	. . 3 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
18		oveq2 7419	. . . . . 6 ⊢ (𝑤 = 0_ℎ → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih 0_ℎ))
19	18	eqeq2d 2780	. . . . 5 ⊢ (𝑤 = 0_ℎ → ((𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
20	19	ralbidv 3194	. . . 4 ⊢ (𝑤 = 0_ℎ → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
21	20	rspcev 3590	. . 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 598	. 2 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
23	6	choccli 31600	. . . 4 ⊢ (⊥‘(null‘𝑇)) ∈ C_ℋ
24	23	chne0i 31746	. . 3 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ ↔ ∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ)
25	23	cheli 31525	. . . . 5 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → 𝑢 ∈ ℋ)
26	3	ffvelcdmi 7079	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑇‘𝑢) ∈ ℂ)
27	26	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑇‘𝑢) ∈ ℂ)
28		hicl 31373	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
29	28	anidms 576	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑢 ·_ih 𝑢) ∈ ℂ)
30	29	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
31		his6 31392	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) = 0 ↔ 𝑢 = 0_ℎ))
32	31	necon3bid 3008	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) ≠ 0 ↔ 𝑢 ≠ 0_ℎ))
33	32	biimpar 482	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ≠ 0)
34	27, 30, 33	divcld 11991	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
35	34	cjcld 15247	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ)
36		simpl 487	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → 𝑢 ∈ ℋ)
37		hvmulcl 31306	. . . . . . . . 9 ⊢ (((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
38	35, 36, 37	syl2anc 595	. . . . . . . 8 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
39	38	adantll 726	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
40		hvmulcl 31306	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
41	26, 40	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
42	3	ffvelcdmi 7079	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ℋ → (𝑇‘𝑣) ∈ ℂ)
43		hvmulcl 31306	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
44	42, 43	sylan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
45	44	ancoms 463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
46		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
47		his2sub 31385	. . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
50		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
51		ax-his3 31377	. . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) = ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)))
53	42	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
54		ax-his3 31377	. . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
56	52, 55	oveq12d 7429	. . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 31310	. . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ)
61	2	lnfnsubi 32339	. . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . 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 32337	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
64	26, 63	sylan 591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
65	2	lnfnmuli 32337	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑣) · (𝑇‘𝑢)))
66		mulcom 11186	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑇‘𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
67	26, 66	sylan2 604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
68	65, 67	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
69	42, 68	sylan 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
70	69	ancoms 463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
71	64, 70	oveq12d 7429	. . . . . . . . . . . . . . . . . 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 11184	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ (𝑇‘𝑣) ∈ ℂ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
73	26, 42, 72	syl2an 607	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
74	73	subidd 11557	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑇‘𝑣)) − ((𝑇‘𝑢) · (𝑇‘𝑣))) = 0)
75	62, 71, 74	3eqtrd 2808	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0)
76		elnlfn 32221	. . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇))
79	6	chssii 31524	. . . . . . . . . . . . . . . . 17 ⊢ (null‘𝑇) ⊆ ℋ
80		ocorth 31584	. . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) ∧ 𝑢 ∈ (⊥‘(null‘𝑇))) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
83	82	ancoms 463	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ (𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ)) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
84	83	anassrs 472	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
85	58, 84	eqtrd 2804	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0)
86		hicl 31373	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
87	86	ancoms 463	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
88	49, 87	mulcld 11229	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
89		mulcl 11184	. . . . . . . . . . . . . . 15 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
90	42, 29, 89	syl2anr 608	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
91	88, 90	subeq0ad 11579	. . . . . . . . . . . . 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 726	. . . . . . . . . . . 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 235	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
94	93	adantlr 727	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
95	88	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
96	42	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
97	30, 33	jca 520	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
98	97	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
99		divmul3 11877	. . . . . . . . . . . 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 1396	. . . . . . . . . . 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 730	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) / (𝑢 ·_ih 𝑢)) = (𝑇‘𝑣))
103	27	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
104	87	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
105		div23 11891	. . . . . . . . . . . 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 1396	. . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
108		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
109		simpll 778	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
110		his52 31380	. . . . . . . . . . . 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 1396	. . . . . . . . . . 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 2807	. . . . . . . . . 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 730	. . . . . . . . 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 2806	. . . . . . . 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 3163	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
116		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
117	116	eqeq2d 2780	. . . . . . . . 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 3194	. . . . . . . 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 3590	. . . . . . 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 595	. . . . . 6 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
121	120	ex 417	. . . . 5 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
122	25, 121	mpdan 699	. . . 4 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
123	122	rexlimiv 3165	. . 3 ⊢ (∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
124	24, 123	sylbi 220	. 2 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
125	22, 124	pm2.61ine 3047	1 ⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ⊆ wss 3913 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 0cc0 11100 · cmul 11105 − cmin 11441 / cdiv 11871 ∗ccj 15147 ℋchba 31212 ·_ℎ csm 31214 ·_ih csp 31215 0_ℎc0v 31217 −_ℎ cmv 31218 ⊥cort 31223 0_ℋc0h 31228 nullcnl 31245 ContFnccnfn 31246 LinFnclf 31247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cc 10419 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179 ax-mulf 11180 ax-hilex 31292 ax-hfvadd 31293 ax-hvcom 31294 ax-hvass 31295 ax-hv0cl 31296 ax-hvaddid 31297 ax-hfvmul 31298 ax-hvmulid 31299 ax-hvmulass 31300 ax-hvdistr1 31301 ax-hvdistr2 31302 ax-hvmul0 31303 ax-hfi 31372 ax-his1 31375 ax-his2 31376 ax-his3 31377 ax-his4 31378 ax-hcompl 31495

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-acn 9928 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-cn 23353 df-cnp 23354 df-lm 23355 df-haus 23441 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cfil 25383 df-cau 25384 df-cmet 25385 df-grpo 30786 df-gid 30787 df-ginv 30788 df-gdiv 30789 df-ablo 30838 df-vc 30852 df-nv 30885 df-va 30888 df-ba 30889 df-sm 30890 df-0v 30891 df-vs 30892 df-nmcv 30893 df-ims 30894 df-dip 30994 df-ssp 31015 df-ph 31106 df-cbn 31156 df-hnorm 31261 df-hba 31262 df-hvsub 31264 df-hlim 31265 df-hcau 31266 df-sh 31500 df-ch 31514 df-oc 31545 df-ch0 31546 df-nlfn 32139 df-cnfn 32140 df-lnfn 32141

This theorem is referenced by: riesz4i 32356 riesz1 32358

W3C validator