Theorem bcled 43335

Description: Inequality for binomial coefficients. (Contributed by metakunt, 12-May-2025.)

Hypotheses

Ref	Expression
bcled.1	⊢ (𝜑 → 𝐴 ∈ ℕ0)
bcled.2	⊢ (𝜑 → 𝐵 ∈ ℕ0)
bcled.3	⊢ (𝜑 → 𝐶 ∈ ℤ)
bcled.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Assertion

Ref	Expression
bcled	⊢ (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))

Proof of Theorem bcled

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcval2 14454	. . . 4 ⊢ (𝐶 ∈ (0...𝐴) → (𝐴C𝐶) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
2	1	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
3		bcled.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
4	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
5	4	faccld 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
6	5	nncnd 12356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
7	4	nn0zd 12723	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
8		bcled.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℤ)
9	8	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
10	7, 9	zsubcld 12813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℤ)
11	9	zred 12808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
12	4	nn0red 12673	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13		0red 11316	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14		elfzle2 13666	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
15	14	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐴)
16	12	recnd 11342	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
17	16	subid1d 11663	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
18	17	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
19	15, 18	breqtrd 5130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
20	11, 12, 13, 19	lesubd 11925	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐴 − 𝐶))
21	10, 20	jca 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐴 − 𝐶)))
22		elnn0z 12711	. . . . . . . . . 10 ⊢ ((𝐴 − 𝐶) ∈ ℕ0 ↔ ((𝐴 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐴 − 𝐶)))
23	21, 22	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℕ0)
24	23	faccld 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℕ)
25	24	nncnd 12356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℂ)
26		elfznn0 13759	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ0)
27	26	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ0)
28	27	faccld 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
29	28	nncnd 12356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
30	24	nnne0d 12393	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ≠ 0)
31	28	nnne0d 12393	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
32	6, 25, 29, 30, 31	divdiv1d 12129	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
33	32	eqcomd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)))
34	5	nnred 12355	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
35	24	nnred 12355	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℝ)
36	34, 35, 30	redivcld 12150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ∈ ℝ)
37		bcled.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℕ0)
38	37	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℕ0)
39	38	faccld 14433	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
40	39	nnred 12355	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
41	38	nn0zd 12723	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
42	41, 9	zsubcld 12813	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
43	38	nn0red 12673	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
44		bcled.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
45	44	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ≤ 𝐵)
46	11, 12, 43, 15, 45	letrd 11472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
47	43	recnd 11342	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
48	47	subid1d 11663	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
49	48	eqcomd 2766	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
50	46, 49	breqtrd 5130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
51	11, 43, 13, 50	lesubd 11925	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵 − 𝐶))
52	42, 51	jca 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((𝐵 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐵 − 𝐶)))
53		elnn0z 12711	. . . . . . . . . . 11 ⊢ ((𝐵 − 𝐶) ∈ ℕ0 ↔ ((𝐵 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐵 − 𝐶)))
54	52, 53	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℕ0)
55	54	faccld 14433	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℕ)
56	55	nnred 12355	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℝ)
57	55	nnne0d 12393	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ≠ 0)
58	40, 56, 57	redivcld 12150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵 − 𝐶))) ∈ ℝ)
59	28	nnrpd 13167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ+)
60		nfv 1946	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ (0...𝐴))
61		fzfid 14121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (0...(𝐶 − 1)) ∈ Fin)
62	12	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ∈ ℝ)
63		elfzelz 13661	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ∈ ℤ)
64	63	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℤ)
65	64	zred 12808	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
66	62, 65	resubcld 11749	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ∈ ℝ)
67		0red 11316	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
68	27	nn0red 12673	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
69	68	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70		1red 11314	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
71	69, 70	resubcld 11749	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
72	62, 67	resubcld 11749	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73		elfzle2 13666	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ≤ (𝐶 − 1))
74	73	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐶 − 1))
75	15	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ≤ 𝐴)
76		0le1 11844	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
77	76	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ 1)
78	69, 67, 62, 70, 75, 77	le2subd 11941	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
79	65, 71, 72, 74, 78	letrd 11472	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
80	65, 62, 67, 79	lesubd 11925	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ (𝐴 − 𝑘))
81	43	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐵 ∈ ℝ)
82	81, 65	resubcld 11749	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐵 − 𝑘) ∈ ℝ)
83	44	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ≤ 𝐵)
84	62, 81, 65, 83	lesub1dd 11937	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ≤ (𝐵 − 𝑘))
85	60, 61, 66, 80, 82, 84	fprodle 16170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
86	4	nn0cnd 12674	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87		fallfacval 16183	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘))
88	86, 27, 87	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘))
89	88	eqcomd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) = (𝐴 FallFac 𝐶))
90	38	nn0cnd 12674	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91		fallfacval 16183	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
92	90, 27, 91	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
93	92	eqcomd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘) = (𝐵 FallFac 𝐶))
94	85, 89, 93	3brtr3d 5135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95		fallfacval4 16216	. . . . . . . . 9 ⊢ (𝐶 ∈ (0...𝐴) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴 − 𝐶))))
96	95	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴 − 𝐶))))
97		0zd 12710	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
98	27	nn0ge0d 12675	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
99	68, 12, 43, 15, 45	letrd 11472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
100	97, 41, 9, 98, 99	elfzd 13652	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101		fallfacval4 16216	. . . . . . . . 9 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
102	100, 101	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
103	94, 96, 102	3brtr3d 5135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
104	36, 58, 59, 103	lediv1dd 13227	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)))
105	39	nncnd 12356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
106	55	nncnd 12356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℂ)
107	105, 106, 29, 57, 31	divdiv1d 12129	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
108	104, 107	breqtrd 5130	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
109	33, 108	eqbrtrd 5126	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
110	37	nn0zd 12723	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
111	110	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112		elfzle1 13665	. . . . . . . 8 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113	112	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
114	3	nn0red 12673	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
115	114	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116	111	zred 12808	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
117	11, 115, 116, 15, 45	letrd 11472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
118	97, 111, 9, 113, 117	elfzd 13652	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119		bcval2 14454	. . . . . 6 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
120	118, 119	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
121	120	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122	109, 121	breqtrd 5130	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
123	2, 122	eqbrtrd 5126	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
124	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
125	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
126		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → ¬ 𝐶 ∈ (0...𝐴))
127		bcval3 14455	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
128	124, 125, 126, 127	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
129		bccl2 14472	. . . . . . 7 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130	129	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131	130	nnnn0d 12672	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ0)
132	131	nn0ge0d 12675	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133		0le0 12449	. . . . . 6 ⊢ 0 ≤ 0
134	133	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ 0)
135	37	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 𝐵 ∈ ℕ0)
136	125	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 𝐶 ∈ ℤ)
137		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → ¬ 𝐶 ∈ (0...𝐵))
138		bcval3 14455	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
139	135, 136, 137, 138	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
140	139	eqcomd 2766	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141	134, 140	breqtrd 5130	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
142	132, 141	pm2.61dan 825	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵C𝐶))
143	128, 142	eqbrtrd 5126	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
144	123, 143	pm2.61dan 825	1 ⊢ (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6541  (class class class)co 7422  ℂcc 11203  ℝcr 11204  0cc0 11205  1c1 11206   · cmul 11210   ≤ cle 11349   − cmin 11546   / cdiv 11978  ℕcn 12340  ℕ₀cn0 12611  ℤcz 12698  ...cfz 13644  !cfa 14422  Ccbc 14451  ∏cprod 16079   FallFac cfallfac 16178

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-inf2 9647  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282  ax-pre-sup 11283

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 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-sup 9439  df-oi 9509  df-card 10025  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-div 11979  df-nn 12341  df-2 12410  df-3 12411  df-n0 12612  df-z 12699  df-uz 12971  df-rp 13126  df-ico 13487  df-fz 13645  df-fzo 13794  df-seq 14150  df-exp 14210  df-fac 14423  df-bc 14452  df-hash 14480  df-cj 15273  df-re 15274  df-im 15275  df-sqrt 15409  df-abs 15410  df-clim 15662  df-prod 16080  df-fallfac 16181

This theorem is used by:  aks6d1c7lem1  43337