lcmineqlem1 - Mathbox for metakunt

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Theorem lcmineqlem1 42030

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 13508	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈ ℂ)
3		ax-1cn 11213	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
4		negsub 11557	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + -𝑥) = (1 − 𝑥))
5	3, 4	mpan 690	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 + -𝑥) = (1 − 𝑥))
6	5	oveq1d 7446	. . . . . . . 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 11508	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ)
9		1cnd 11256	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
10		lcmineqlem1.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
11		lcmineqlem1.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
12	11	nnnn0d 12587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
13		lcmineqlem1.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
14	13	nnnn0d 12587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
15		nn0sub 12576	. . . . . . . . . . . 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 15866	. . . . . . . . . . . 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 1127	. . . . . . . . . . 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 1122	. . . . . . . . . 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 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))
25		elfzelz 13564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℤ)
26	13	nnzd 12640	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
27	11	nnzd 12640	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		zsubcl 12659	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ)
29	26, 27, 28	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℤ)
30		zsubcl 12659	. . . . . . . . . . . . . . . . . . 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 14132	. . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑((𝑁 − 𝑀) − 𝑘)) = 1)
36	35	oveq1d 7446	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (1 · (-𝑥↑𝑘)))
37	8	3ad2ant2 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → -𝑥 ∈ ℂ)
38		elfznn0 13660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ₀)
39	38	3ad2ant3 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ₀)
40		expcl 14120	. . . . . . . . . . . . . . . 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 11279	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1 · (-𝑥↑𝑘)) = (-𝑥↑𝑘))
43	36, 42	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (-𝑥↑𝑘))
44		mulm1 11704	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (-1 · 𝑥) = -𝑥)
45	44	oveq1d 7446	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((-1 · 𝑥)↑𝑘) = (-𝑥↑𝑘))
46	45	3ad2ant2 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = (-𝑥↑𝑘))
47	43, 46	eqtr4d 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1 · 𝑥)↑𝑘))
48		neg1cn 12380	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
49		mulexp 14142	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘)))
53	47, 52	eqtrd 2777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1↑𝑘) · (𝑥↑𝑘)))
54	53	oveq2d 7447	. . . . . . . . . 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 14361	. . . . . . . . . . . . . 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 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
58	57	nn0cnd 12589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℂ)
59		expcl 14120	. . . . . . . . . . . 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 14120	. . . . . . . . . . . . 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 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ)
64	58, 60, 63	mulassd 11284	. . . . . . . . . 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 2780	. . . . . . . . 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 11282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)))
67	66	oveq1d 7446	. . . . . . . . 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 2777	. . . . . . . 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 1119	. . . . . . 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 15738	. . . . . 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 2777	. . . . 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 7447	. . 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 25817	. 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 2789	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 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 -cneg 11493 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 [,]cicc 13390 ...cfz 13547 ↑cexp 14102 Ccbc 14341 Σcsu 15722 ∫citg 25653

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-itg 25658

This theorem is referenced by: lcmineqlem2 42031

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