knoppndvlem6 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem6 35696

Description: Lemma for knoppndv 35713. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

**Hypotheses**
Ref	Expression
knoppndvlem6.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
knoppndvlem6.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
knoppndvlem6.w	⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
knoppndvlem6.a	⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)
knoppndvlem6.c	⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
knoppndvlem6.j	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
knoppndvlem6.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
knoppndvlem6.n	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
knoppndvlem6	⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))

Distinct variable groups: 𝐴,𝑖,𝑛,𝑤,𝑦 𝑥,𝐴,𝑖,𝑤 𝐶,𝑛,𝑦 𝑖,𝐹,𝑤 𝑖,𝐽,𝑛,𝑦 𝑛,𝑁,𝑦 𝑥,𝑁 𝑇,𝑛,𝑦 𝜑,𝑖,𝑛,𝑤,𝑦 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥,𝑤,𝑖) 𝑇(𝑥,𝑤,𝑖) 𝐹(𝑥,𝑦,𝑛) 𝐽(𝑥,𝑤) 𝑀(𝑥,𝑦,𝑤,𝑖,𝑛) 𝑁(𝑤,𝑖) 𝑊(𝑥,𝑦,𝑤,𝑖,𝑛)

**Proof of Theorem knoppndvlem6**
Step	Hyp	Ref	Expression
1		knoppndvlem6.w	. . . 4 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
2		fveq2 6890	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴))
3	2	fveq1d 6892	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑤)‘𝑖) = ((𝐹‘𝐴)‘𝑖))
4	3	sumeq2sdv 15654	. . . 4 ⊢ (𝑤 = 𝐴 → Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖) = Σ𝑖 ∈ ℕ₀ ((𝐹‘𝐴)‘𝑖))
5		knoppndvlem6.a	. . . . . 6 ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
7		knoppndvlem6.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8		knoppndvlem6.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
9	8	nn0zd 12588	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
10		knoppndvlem6.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
11	7, 9, 10	knoppndvlem1 35691	. . . . 5 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
12	6, 11	eqeltrd 2831	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13		sumex 15638	. . . . 5 ⊢ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝐴)‘𝑖) ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ℕ₀ ((𝐹‘𝐴)‘𝑖) ∈ V)
15	1, 4, 12, 14	fvmptd3 7020	. . 3 ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ ℕ₀ ((𝐹‘𝐴)‘𝑖))
16		nn0uz 12868	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
17		eqid 2730	. . . 4 ⊢ (ℤ_≥‘(𝐽 + 1)) = (ℤ_≥‘(𝐽 + 1))
18		peano2nn0 12516	. . . . 5 ⊢ (𝐽 ∈ ℕ₀ → (𝐽 + 1) ∈ ℕ₀)
19	8, 18	syl 17	. . . 4 ⊢ (𝜑 → (𝐽 + 1) ∈ ℕ₀)
20		eqidd 2731	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐹‘𝐴)‘𝑖) = ((𝐹‘𝐴)‘𝑖))
21		knoppndvlem6.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
22		knoppndvlem6.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
23	7	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑁 ∈ ℕ)
24		knoppndvlem6.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
25	24	knoppndvlem3 35693	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
26	25	simpld 493	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
27	26	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝐶 ∈ ℝ)
28	12	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝐴 ∈ ℝ)
29		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
30	21, 22, 23, 27, 28, 29	knoppcnlem3 35674	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐹‘𝐴)‘𝑖) ∈ ℝ)
31	30	recnd 11246	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝐹‘𝐴)‘𝑖) ∈ ℂ)
32	21, 22, 1, 12, 24, 7	knoppndvlem4 35694	. . . . 5 ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴))
33		seqex 13972	. . . . . 6 ⊢ seq0( + , (𝐹‘𝐴)) ∈ V
34		fvex 6903	. . . . . 6 ⊢ (𝑊‘𝐴) ∈ V
35	33, 34	breldm 5907	. . . . 5 ⊢ (seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴) → seq0( + , (𝐹‘𝐴)) ∈ dom ⇝ )
36	32, 35	syl 17	. . . 4 ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ∈ dom ⇝ )
37	16, 17, 19, 20, 31, 36	isumsplit 15790	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ ℕ₀ ((𝐹‘𝐴)‘𝑖) = (Σ𝑖 ∈ (0...((𝐽 + 1) − 1))((𝐹‘𝐴)‘𝑖) + Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖)))
38	8	nn0cnd 12538	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℂ)
39		1cnd 11213	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
40	38, 39	pncand 11576	. . . . . 6 ⊢ (𝜑 → ((𝐽 + 1) − 1) = 𝐽)
41	40	oveq2d 7427	. . . . 5 ⊢ (𝜑 → (0...((𝐽 + 1) − 1)) = (0...𝐽))
42	41	sumeq1d 15651	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ (0...((𝐽 + 1) − 1))((𝐹‘𝐴)‘𝑖) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
43	42	oveq1d 7426	. . 3 ⊢ (𝜑 → (Σ𝑖 ∈ (0...((𝐽 + 1) − 1))((𝐹‘𝐴)‘𝑖) + Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖)) = (Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) + Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖)))
44	15, 37, 43	3eqtrd 2774	. 2 ⊢ (𝜑 → (𝑊‘𝐴) = (Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) + Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖)))
45	12	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝐴 ∈ ℝ)
46		eluznn0 12905	. . . . . . . . 9 ⊢ (((𝐽 + 1) ∈ ℕ₀ ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝑖 ∈ ℕ₀)
47	19, 46	sylan 578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝑖 ∈ ℕ₀)
48	22, 45, 47	knoppcnlem1 35672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → ((𝐹‘𝐴)‘𝑖) = ((𝐶↑𝑖) · (𝑇‘(((2 · 𝑁)↑𝑖) · 𝐴))))
49	5	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
50	49	oveq2d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (((2 · 𝑁)↑𝑖) · 𝐴) = (((2 · 𝑁)↑𝑖) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)))
51	7	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝑁 ∈ ℕ)
52	47	nn0zd 12588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝑖 ∈ ℤ)
53	9	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝐽 ∈ ℤ)
54	10	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝑀 ∈ ℤ)
55		eluzle 12839	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (ℤ_≥‘(𝐽 + 1)) → (𝐽 + 1) ≤ 𝑖)
56	55	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (𝐽 + 1) ≤ 𝑖)
57	53, 52	jca 510	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (𝐽 ∈ ℤ ∧ 𝑖 ∈ ℤ))
58		zltp1le 12616	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝐽 < 𝑖 ↔ (𝐽 + 1) ≤ 𝑖))
59	57, 58	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (𝐽 < 𝑖 ↔ (𝐽 + 1) ≤ 𝑖))
60	56, 59	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝐽 < 𝑖)
61	51, 52, 53, 54, 60	knoppndvlem2 35692	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (((2 · 𝑁)↑𝑖) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ)
62	50, 61	eqeltrd 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (((2 · 𝑁)↑𝑖) · 𝐴) ∈ ℤ)
63	21, 62	dnizeq0 35654	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (𝑇‘(((2 · 𝑁)↑𝑖) · 𝐴)) = 0)
64	63	oveq2d 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → ((𝐶↑𝑖) · (𝑇‘(((2 · 𝑁)↑𝑖) · 𝐴))) = ((𝐶↑𝑖) · 0))
65	26	recnd 11246	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
66	65	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → 𝐶 ∈ ℂ)
67	66, 47	expcld 14115	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → (𝐶↑𝑖) ∈ ℂ)
68	67	mul01d 11417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → ((𝐶↑𝑖) · 0) = 0)
69	48, 64, 68	3eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘(𝐽 + 1))) → ((𝐹‘𝐴)‘𝑖) = 0)
70	69	sumeq2dv 15653	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖) = Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))0)
71		ssidd 4004	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘(𝐽 + 1)) ⊆ (ℤ_≥‘(𝐽 + 1)))
72	71	orcd 869	. . . . . 6 ⊢ (𝜑 → ((ℤ_≥‘(𝐽 + 1)) ⊆ (ℤ_≥‘(𝐽 + 1)) ∨ (ℤ_≥‘(𝐽 + 1)) ∈ Fin))
73		sumz 15672	. . . . . 6 ⊢ (((ℤ_≥‘(𝐽 + 1)) ⊆ (ℤ_≥‘(𝐽 + 1)) ∨ (ℤ_≥‘(𝐽 + 1)) ∈ Fin) → Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))0 = 0)
74	72, 73	syl 17	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))0 = 0)
75	70, 74	eqtrd 2770	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖) = 0)
76	75	oveq2d 7427	. . 3 ⊢ (𝜑 → (Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) + Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖)) = (Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) + 0))
77	21, 22, 12, 26, 7	knoppndvlem5 35695	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ)
78	77	recnd 11246	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℂ)
79	78	addridd 11418	. . 3 ⊢ (𝜑 → (Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) + 0) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
80	76, 79	eqtrd 2770	. 2 ⊢ (𝜑 → (Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) + Σ𝑖 ∈ (ℤ_≥‘(𝐽 + 1))((𝐹‘𝐴)‘𝑖)) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
81	44, 80	eqtrd 2770	1 ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 (,)cioo 13328 ...cfz 13488 ⌊cfl 13759 seqcseq 13970 ↑cexp 14031 abscabs 15185 ⇝ cli 15432 Σcsu 15636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-ioo 13332 df-ico 13334 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ulm 26125

This theorem is referenced by: knoppndvlem15 35705

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