riesz3i - Hilbert Space Explorer

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Theorem riesz3i 32487

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 31428	. . 3 ⊢ 0_ℎ ∈ ℋ
2		nlelch.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfnfi 32466	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
4		fveq2 6885	. . . . . . . . 9 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (⊥‘(⊥‘(null‘𝑇))) = (⊥‘0_ℋ))
5		nlelch.2	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContFn
6	2, 5	nlelchi 32486	. . . . . . . . . 10 ⊢ (null‘𝑇) ∈ C_ℋ
7	6	ococi 31830	. . . . . . . . 9 ⊢ (⊥‘(⊥‘(null‘𝑇))) = (null‘𝑇)
8		choc0 31751	. . . . . . . . 9 ⊢ (⊥‘0_ℋ) = ℋ
9	4, 7, 8	3eqtr3g 2823	. . . . . . . 8 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → (null‘𝑇) = ℋ)
10	9	eleq2d 2851	. . . . . . 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 32354	. . . . . 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 31522	. . . . . 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 2803	. . . 4 ⊢ (((⊥‘(null‘𝑇)) = 0_ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
17	16	ralrimiva 3159	. . 3 ⊢ ((⊥‘(null‘𝑇)) = 0_ℋ → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ))
18		oveq2 7427	. . . . . 6 ⊢ (𝑤 = 0_ℎ → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih 0_ℎ))
19	18	eqeq2d 2776	. . . . 5 ⊢ (𝑤 = 0_ℎ → ((𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
20	19	ralbidv 3190	. . . 4 ⊢ (𝑤 = 0_ℎ → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 𝑤) ↔ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih 0_ℎ)))
21	20	rspcev 3583	. . 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 31732	. . . 4 ⊢ (⊥‘(null‘𝑇)) ∈ C_ℋ
24	23	chne0i 31878	. . 3 ⊢ ((⊥‘(null‘𝑇)) ≠ 0_ℋ ↔ ∃𝑢 ∈ (⊥‘(null‘𝑇))𝑢 ≠ 0_ℎ)
25	23	cheli 31657	. . . . 5 ⊢ (𝑢 ∈ (⊥‘(null‘𝑇)) → 𝑢 ∈ ℋ)
26	3	ffvelcdmi 7082	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℋ → (𝑇‘𝑢) ∈ ℂ)
27	26	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → (𝑇‘𝑢) ∈ ℂ)
28		hicl 31505	. . . . . . . . . . . . 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 31524	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℋ → ((𝑢 ·_ih 𝑢) = 0 ↔ 𝑢 = 0_ℎ))
32	31	necon3bid 3004	. . . . . . . . . . . 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 12008	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℋ ∧ 𝑢 ≠ 0_ℎ) → ((𝑇‘𝑢) / (𝑢 ·_ih 𝑢)) ∈ ℂ)
35	34	cjcld 15273	. . . . . . . . 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 31438	. . . . . . . . 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 31438	. . . . . . . . . . . . . . . . 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 7082	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ℋ → (𝑇‘𝑣) ∈ ℂ)
43		hvmulcl 31438	. . . . . . . . . . . . . . . . . 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 31517	. . . . . . . . . . . . . . . 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 31509	. . . . . . . . . . . . . . . . 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 31509	. . . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . 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 31442	. . . . . . . . . . . . . . . . . 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 32471	. . . . . . . . . . . . . . . . . . 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 32469	. . . . . . . . . . . . . . . . . . . 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 32469	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇‘𝑣) ∈ ℂ ∧ 𝑢 ∈ ℋ) → (𝑇‘((𝑇‘𝑣) ·_ℎ 𝑢)) = ((𝑇‘𝑣) · (𝑇‘𝑢)))
66		mulcom 11203	. . . . . . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . . . 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 11201	. . . . . . . . . . . . . . . . . . . 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 11574	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑇‘𝑣)) − ((𝑇‘𝑢) · (𝑇‘𝑣))) = 0)
75	62, 71, 74	3eqtrd 2804	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → (𝑇‘(((𝑇‘𝑢) ·_ℎ 𝑣) −_ℎ ((𝑇‘𝑣) ·_ℎ 𝑢))) = 0)
76		elnlfn 32353	. . . . . . . . . . . . . . . . . 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 31656	. . . . . . . . . . . . . . . . 17 ⊢ (null‘𝑇) ⊆ ℋ
80		ocorth 31716	. . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑣 ∈ ℋ) → (((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) − ((𝑇‘𝑣) · (𝑢 ·_ih 𝑢))) = 0)
86		hicl 31505	. . . . . . . . . . . . . . . 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 11246	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ) → ((𝑇‘𝑢) · (𝑣 ·_ih 𝑢)) ∈ ℂ)
89		mulcl 11201	. . . . . . . . . . . . . . 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 11596	. . . . . . . . . . . . 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 11894	. . . . . . . . . . . 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 11908	. . . . . . . . . . . 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 31512	. . . . . . . . . . . 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 2803	. . . . . . . . . 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 2802	. . . . . . . 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 3159	. . . . . . 7 ⊢ (((𝑢 ∈ (⊥‘(null‘𝑇)) ∧ 𝑢 ∈ ℋ) ∧ 𝑢 ≠ 0_ℎ) → ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
116		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑤 = ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢) → (𝑣 ·_ih 𝑤) = (𝑣 ·_ih ((∗‘((𝑇‘𝑢) / (𝑢 ·_ih 𝑢))) ·_ℎ 𝑢)))
117	116	eqeq2d 2776	. . . . . . . . 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 3190	. . . . . . . 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 3583	. . . . . . 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 3161	. . 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 3043	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 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 ⊆ wss 3906 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11115 0cc0 11117 · cmul 11122 − cmin 11458 / cdiv 11888 ∗ccj 15173 ℋchba 31344 ·_ℎ csm 31346 ·_ih csp 31347 0_ℎc0v 31349 −_ℎ cmv 31350 ⊥cort 31355 0_ℋc0h 31360 nullcnl 31377 ContFnccnfn 31378 LinFnclf 31379

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cc 10434 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196 ax-mulf 11197 ax-hilex 31424 ax-hfvadd 31425 ax-hvcom 31426 ax-hvass 31427 ax-hv0cl 31428 ax-hvaddid 31429 ax-hfvmul 31430 ax-hvmulid 31431 ax-hvmulass 31432 ax-hvdistr1 31433 ax-hvdistr2 31434 ax-hvmul0 31435 ax-hfi 31504 ax-his1 31507 ax-his2 31508 ax-his3 31509 ax-his4 31510 ax-hcompl 31627

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-acn 9944 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-sum 15764 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-pt 17521 df-prds 17524 df-xrs 17580 df-qtop 17585 df-imas 17586 df-xps 17588 df-mre 17662 df-mrc 17663 df-acs 17665 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-mulg 19180 df-cntz 19433 df-cmn 19898 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-fbas 21571 df-fg 21572 df-cnfld 21575 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-cld 23228 df-ntr 23229 df-cls 23230 df-nei 23307 df-cn 23436 df-cnp 23437 df-lm 23438 df-haus 23524 df-tx 23772 df-hmeo 23965 df-fil 24056 df-fm 24148 df-flim 24149 df-flf 24150 df-xms 24530 df-ms 24531 df-tms 24532 df-cfil 25467 df-cau 25468 df-cmet 25469 df-grpo 30918 df-gid 30919 df-ginv 30920 df-gdiv 30921 df-ablo 30970 df-vc 30984 df-nv 31017 df-va 31020 df-ba 31021 df-sm 31022 df-0v 31023 df-vs 31024 df-nmcv 31025 df-ims 31026 df-dip 31126 df-ssp 31147 df-ph 31238 df-cbn 31288 df-hnorm 31393 df-hba 31394 df-hvsub 31396 df-hlim 31397 df-hcau 31398 df-sh 31632 df-ch 31646 df-oc 31677 df-ch0 31678 df-nlfn 32271 df-cnfn 32272 df-lnfn 32273

This theorem is used by: riesz4i 32488 riesz1 32490

W3C validator