lcmineqlem1 - Mathbox for metakunt

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Theorem lcmineqlem1 42142

Description: Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.)

**Hypotheses**
Ref	Expression
lcmineqlem1.1	⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥
lcmineqlem1.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
lcmineqlem1.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
lcmineqlem1.4	⊢ (𝜑 → 𝑀 ≤ 𝑁)

**Assertion**
Ref	Expression
lcmineqlem1	⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥)

Distinct variable groups: 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘,𝑥 Allowed substitution hints: 𝐹(𝑥,𝑘) 𝑀(𝑥) 𝑁(𝑥)

**Proof of Theorem lcmineqlem1**
Step	Hyp	Ref	Expression
1		lcmineqlem1.1	. 2 ⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥
2		elunitcn 13370	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ)
3		ax-1cn 11071	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
4		negsub 11416	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + -𝑥) = (1 − 𝑥))
5	3, 4	mpan 690	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 + -𝑥) = (1 − 𝑥))
6	5	oveq1d 7367	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀)))
7	6	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀)))
8		negcl 11367	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ)
9		1cnd 11114	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
10		lcmineqlem1.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
11		lcmineqlem1.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
12	11	nnnn0d 12449	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
13		lcmineqlem1.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
14	13	nnnn0d 12449	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
15		nn0sub 12438	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ₀))
16	12, 14, 15	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ₀))
17	10, 16	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
18		binom 15739	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ -𝑥 ∈ ℂ ∧ (𝑁 − 𝑀) ∈ ℕ₀) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))
19	18	3com23 1126	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (𝑁 − 𝑀) ∈ ℕ₀ ∧ -𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))
20	19	3expia 1121	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑁 − 𝑀) ∈ ℕ₀) → (-𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))))
21	9, 17, 20	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (-𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))))
22	8, 21	syl5 34	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))))
23	22	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))
24	7, 23	eqtr3d 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))
25		elfzelz 13426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℤ)
26	13	nnzd 12501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
27	11	nnzd 12501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		zsubcl 12520	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ)
29	26, 27, 28	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℤ)
30		zsubcl 12520	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 − 𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ)
31	29, 30	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ)
32	25, 31	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ)
33		1exp 14000	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 − 𝑀) − 𝑘) ∈ ℤ → (1↑((𝑁 − 𝑀) − 𝑘)) = 1)
34	32, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑((𝑁 − 𝑀) − 𝑘)) = 1)
35	34	3adant2 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑((𝑁 − 𝑀) − 𝑘)) = 1)
36	35	oveq1d 7367	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (1 · (-𝑥↑𝑘)))
37	8	3ad2ant2 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → -𝑥 ∈ ℂ)
38		elfznn0 13522	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ₀)
39	38	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ₀)
40		expcl 13988	. . . . . . . . . . . . . . . 16 ⊢ ((-𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-𝑥↑𝑘) ∈ ℂ)
41	37, 39, 40	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-𝑥↑𝑘) ∈ ℂ)
42	41	mullidd 11137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1 · (-𝑥↑𝑘)) = (-𝑥↑𝑘))
43	36, 42	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (-𝑥↑𝑘))
44		mulm1 11565	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (-1 · 𝑥) = -𝑥)
45	44	oveq1d 7367	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((-1 · 𝑥)↑𝑘) = (-𝑥↑𝑘))
46	45	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = (-𝑥↑𝑘))
47	43, 46	eqtr4d 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1 · 𝑥)↑𝑘))
48		neg1cn 12117	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
49		mulexp 14010	. . . . . . . . . . . . . . 15 ⊢ ((-1 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘)))
50	48, 49	mp3an1 1450	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘)))
51	38, 50	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘)))
52	51	3adant1 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘)))
53	47, 52	eqtrd 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1↑𝑘) · (𝑥↑𝑘)))
54	53	oveq2d 7368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = (((𝑁 − 𝑀)C𝑘) · ((-1↑𝑘) · (𝑥↑𝑘))))
55		bccl 14231	. . . . . . . . . . . . . 14 ⊢ (((𝑁 − 𝑀) ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
56	17, 25, 55	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
57	56	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
58	57	nn0cnd 12451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℂ)
59		expcl 13988	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
60	48, 39, 59	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-1↑𝑘) ∈ ℂ)
61		expcl 13988	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
62	38, 61	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ)
63	62	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ)
64	58, 60, 63	mulassd 11142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) · (𝑥↑𝑘)) = (((𝑁 − 𝑀)C𝑘) · ((-1↑𝑘) · (𝑥↑𝑘))))
65	54, 64	eqtr4d 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = ((((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) · (𝑥↑𝑘)))
66	58, 60	mulcomd 11140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)))
67	66	oveq1d 7367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) · (𝑥↑𝑘)) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))
68	65, 67	eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))
69	68	3expa 1118	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))
70	69	sumeq2dv 15611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))
71	24, 70	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))
72	2, 71	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))
73	72	oveq2d 7368	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) = ((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))))
74	73	itgeq2dv 25711	. 2 ⊢ (𝜑 → ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥)
75	1, 74	eqtrid 2780	1 ⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 class class class wbr 5093 (class class class)co 7352 ℂcc 11011 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 ≤ cle 11154 − cmin 11351 -cneg 11352 ℕcn 12132 ℕ₀cn0 12388 ℤcz 12475 [,]cicc 13250 ...cfz 13409 ↑cexp 13970 Ccbc 14211 Σcsu 15595 ∫citg 25547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-icc 13254 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-fac 14183 df-bc 14212 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-clim 15397 df-sum 15596 df-itg 25552

This theorem is referenced by: lcmineqlem2 42143

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