riesz3i - Hilbert Space Explorer

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Theorem riesz3i 32676

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 31617	. . 3 ⊢ 0_ℎ ∈ ℋ
2		nlelch.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfnfi 32655	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
4		fveq2 6893	. . . . . . . . 9 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (⊥‘(⊥‘(null‘𝑇))) = (⊥‘0_ℋ))
5		nlelch.2	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContFn
6	2, 5	nlelchi 32675	. . . . . . . . . 10 ⊢ (null‘𝑇) ∈ C_ℋ
7	6	ococi 32019	. . . . . . . . 9 ⊢ (⊥‘(⊥‘(null‘𝑇))) = (null‘𝑇)
8		choc0 31940	. . . . . . . . 9 ⊢ (⊥‘0_ℋ) = ℋ
9	4, 7, 8	3eqtr3g 2821	. . . . . . . 8 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (null‘𝑇) = ℋ)
10	9	eleq2d 2849	. . . . . . 7 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (𝑣 ∈ (null‘𝑇) ↔ 𝑣 ∈ ℋ))
11	10	biimpar 483	. . . . . 6 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ (null‘𝑇))
12		elnlfn2 32543	. . . . . 6 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝑣 ∈ (null‘𝑇)) → (𝑇‘𝑣) = 0)
13	3, 11, 12	sylancr 599	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = 0)
14		hi02 31711	. . . . . 6 ⊢ (𝑣 ∈ ℋ → (𝑣 ·_ih 0_ℎ) = 0)
15	14	adantl 487	. . . . 5 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 0_ℎ) = 0)
16	13, 15	eqtr4d 2801	. . . 4 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
17	16	ralrimiva 3157	. . 3 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
18		oveq2 7435	. . . . . 6 ⊢ (𝑤 = 0_ℎ → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih 0_ℎ))
19	18	eqeq2d 2774	. . . . 5 ⊢ (𝑤 = 0_ℎ → ((𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
20	19	ralbidv 3188	. . . 4 ⊢ (𝑤 = 0_ℎ → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
21	20	rspcev 3579	. . 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 599	. 2 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
23	6	choccli 31921	. . . 4 ⊢ (⊥‘(null‘𝑇)) ∈ C_ℋ
24	23	chne0i 32067	. . 3 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ ↔ ∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ)
25	23	cheli 31846	. . . . 5 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → 𝑢 ∈ ℋ)
26	3	ffvelcdmi 7091	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑇‘𝑢) ∈ ℂ)
27	26	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑇‘𝑢) ∈ ℂ)
28		hicl 31694	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
29	28	anidms 577	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑢 ·_ih 𝑢) ∈ ℂ)
30	29	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ∈ ℂ)
31		his6 31713	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) = 0 ↔ 𝑢 = 0_ℎ))
32	31	necon3bid 3002	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) ≠ 0 ↔ 𝑢 ≠ 0_ℎ))
33	32	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑢 ·_ih 𝑢) ≠ 0)
34	27, 30, 33	divcld 12103	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
35	34	cjcld 15375	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ)
36		simpl 488	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → 𝑢 ∈ ℋ)
37		hvmulcl 31627	. . . . . . . . 9 ⊢ (((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
38	35, 36, 37	syl2anc 596	. . . . . . . 8 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
39	38	adantll 727	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) ∈ ℋ)
40		hvmulcl 31627	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
41	26, 40	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) ·_ℎ 𝑣) ∈ ℋ)
42	3	ffvelcdmi 7091	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ℋ → (𝑇‘𝑣) ∈ ℂ)
43		hvmulcl 31627	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
44	42, 43	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
45	44	ancoms 464	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) ·_ℎ 𝑢) ∈ ℋ)
46		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
47		his2sub 31706	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
50		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
51		ax-his3 31698	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) ·_ih 𝑢) = ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)))
53	42	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
54		ax-his3 31698	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑣) ·_ℎ 𝑢) ·_ih 𝑢) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
56	52, 55	oveq12d 7445	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 31631	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ ℋ)
61	2	lnfnsubi 32660	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 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 32658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
64	26, 63	sylan 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑢) ·_ℎ 𝑣)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
65	2	lnfnmuli 32658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑣) · (𝑇‘𝑢)))
66		mulcom 11296	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑇‘𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
67	26, 66	sylan2 605	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → ((𝑇‘𝑣) · (𝑇‘𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
68	65, 67	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
69	42, 68	sylan 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
70	69	ancoms 464	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑢) · (𝑇‘𝑣)))
71	64, 70	oveq12d 7445	. . . . . . . . . . . . . . . . . 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 11294	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇‘𝑢) ∈ ℂ ∧ (𝑇‘𝑣) ∈ ℂ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
73	26, 42, 72	syl2an 608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑇‘𝑣)) ∈ ℂ)
74	73	subidd 11667	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑇‘𝑣)) − ((𝑇‘𝑢) · (𝑇‘𝑣))) = 0)
75	62, 71, 74	3eqtrd 2802	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0)
76		elnlfn 32542	. . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ∈ (null‘𝑇))
79	6	chssii 31845	. . . . . . . . . . . . . . . . 17 ⊢ (null‘𝑇) ⊆ ℋ
80		ocorth 31905	. . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) ∧ 𝑢 ∈ (⊥‘(null‘𝑇))) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
83	82	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ (𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ)) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
84	83	anassrs 473	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → ((((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢)) ·_ih 𝑢) = 0)
85	58, 84	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0)
86		hicl 31694	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ℋ ∧ 𝑢 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
87	86	ancoms 464	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
88	49, 87	mulcld 11339	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
89		mulcl 11294	. . . . . . . . . . . . . . 15 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ∈ ℂ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
90	42, 29, 89	syl2anr 609	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)) ∈ ℂ)
91	88, 90	subeq0ad 11687	. . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 728	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) = ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢)))
95	88	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
96	42	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) ∈ ℂ)
97	30, 33	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
98	97	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑢 ·_ih 𝑢) ∈ ℂ ∧ (𝑢 ·_ih 𝑢) ≠ 0))
99		divmul3 11989	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 731	. . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑢) ∈ ℂ)
104	87	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → (𝑣 ·_ih 𝑢) ∈ ℂ)
105		div23 12003	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
108		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑣 ∈ ℋ)
109		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) ∧ 𝑣 ∈ ℋ) → 𝑢 ∈ ℋ)
110		his52 31701	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 2801	. . . . . . . . . 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 731	. . . . . . . . 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 2800	. . . . . . . 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 3157	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
116		oveq2 7435	. . . . . . . . . 10 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
117	116	eqeq2d 2774	. . . . . . . . 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 3188	. . . . . . . 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 3579	. . . . . . 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 596	. . . . . 6 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤))
121	120	ex 418	. . . . 5 ⊢ ((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
122	25, 121	mpdan 700	. . . 4 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → (𝑢 ≠ 0_ℎ → ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)))
123	122	rexlimiv 3159	. . 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 3041	1 ⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ⊆ wss 3899 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 0cc0 11210 · cmul 11215 − cmin 11551 / cdiv 11983 ∗ccj 15275 ℋchba 31533 ·_ℎ csm 31535 ·_ih csp 31536 0_ℎc0v 31538 −_ℎ cmv 31539 ⊥cort 31544 0_ℋc0h 31549 nullcnl 31566 ContFnccnfn 31567 LinFnclf 31568

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cc 10523 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289 ax-mulf 11290 ax-hilex 31613 ax-hfvadd 31614 ax-hvcom 31615 ax-hvass 31616 ax-hv0cl 31617 ax-hvaddid 31618 ax-hfvmul 31619 ax-hvmulid 31620 ax-hvmulass 31621 ax-hvdistr1 31622 ax-hvdistr2 31623 ax-hvmul0 31624 ax-hfi 31693 ax-his1 31696 ax-his2 31697 ax-his3 31698 ax-his4 31699 ax-hcompl 31816

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-supp 8188 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8728 df-map 8860 df-pm 8861 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9365 df-fi 9413 df-sup 9444 df-inf 9445 df-oi 9514 df-card 10030 df-acn 10033 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-xneg 13251 df-xadd 13252 df-xmul 13253 df-ioo 13490 df-ico 13492 df-icc 13493 df-fz 13650 df-fzo 13799 df-fl 13942 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-clim 15667 df-rlim 15668 df-sum 15866 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-sca 17456 df-vsca 17457 df-ip 17458 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-hom 17464 df-cco 17465 df-rest 17605 df-topn 17606 df-0g 17624 df-gsum 17625 df-topgen 17626 df-pt 17627 df-prds 17630 df-xrs 17686 df-qtop 17691 df-imas 17692 df-xps 17694 df-mre 17768 df-mrc 17769 df-acs 17771 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-mulg 19290 df-cntz 19543 df-cmn 20008 df-psmet 21682 df-xmet 21683 df-met 21684 df-bl 21685 df-mopn 21686 df-fbas 21687 df-fg 21688 df-cnfld 21691 df-top 23224 df-topon 23241 df-topsp 23263 df-bases 23276 df-cld 23349 df-ntr 23350 df-cls 23351 df-nei 23428 df-cn 23557 df-cnp 23558 df-lm 23559 df-haus 23645 df-tx 23893 df-hmeo 24086 df-fil 24177 df-fm 24269 df-flim 24270 df-flf 24271 df-xms 24651 df-ms 24652 df-tms 24653 df-cfil 25588 df-cau 25589 df-cmet 25590 df-grpo 31107 df-gid 31108 df-ginv 31109 df-gdiv 31110 df-ablo 31159 df-vc 31173 df-nv 31206 df-va 31209 df-ba 31210 df-sm 31211 df-0v 31212 df-vs 31213 df-nmcv 31214 df-ims 31215 df-dip 31315 df-ssp 31336 df-ph 31427 df-cbn 31477 df-hnorm 31582 df-hba 31583 df-hvsub 31585 df-hlim 31586 df-hcau 31587 df-sh 31821 df-ch 31835 df-oc 31866 df-ch0 31867 df-nlfn 32460 df-cnfn 32461 df-lnfn 32462

This theorem is used by: riesz4i 32677 riesz1 32679

W3C validator