Theorem bcled 43494

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 14490	. . . 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 14469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
6	5	nncnd 12391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
7	4	nn0zd 12758	. . . . . . . . . . . 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 12848	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℤ)
11	9	zred 12843	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
12	4	nn0red 12708	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13		0red 11350	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14		elfzle2 13701	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
15	14	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐴)
16	12	recnd 11376	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
17	16	subid1d 11697	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
18	17	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
19	15, 18	breqtrd 5137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
20	11, 12, 13, 19	lesubd 11960	. . . . . . . . . . 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 12746	. . . . . . . . . 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 14469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℕ)
25	24	nncnd 12391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℂ)
26		elfznn0 13794	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ0)
27	26	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ0)
28	27	faccld 14469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
29	28	nncnd 12391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
30	24	nnne0d 12428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ≠ 0)
31	28	nnne0d 12428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
32	6, 25, 29, 30, 31	divdiv1d 12164	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
33	32	eqcomd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)))
34	5	nnred 12390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
35	24	nnred 12390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℝ)
36	34, 35, 30	redivcld 12185	. . . . . . 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 14469	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
40	39	nnred 12390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
41	38	nn0zd 12758	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
42	41, 9	zsubcld 12848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
43	38	nn0red 12708	. . . . . . . . . . . . 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 11506	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
47	43	recnd 11376	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
48	47	subid1d 11697	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
49	48	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
50	46, 49	breqtrd 5137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
51	11, 43, 13, 50	lesubd 11960	. . . . . . . . . . . 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 12746	. . . . . . . . . . 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 14469	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℕ)
56	55	nnred 12390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℝ)
57	55	nnne0d 12428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ≠ 0)
58	40, 56, 57	redivcld 12185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵 − 𝐶))) ∈ ℝ)
59	28	nnrpd 13202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ+)
60		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ (0...𝐴))
61		fzfid 14156	. . . . . . . . . 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 13696	. . . . . . . . . . . . 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 12843	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
66	62, 65	resubcld 11783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ∈ ℝ)
67		0red 11350	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
68	27	nn0red 12708	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
69	68	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70		1red 11348	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
71	69, 70	resubcld 11783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
72	62, 67	resubcld 11783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73		elfzle2 13701	. . . . . . . . . . . . 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 11879	. . . . . . . . . . . . . 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 11976	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
79	65, 71, 72, 74, 78	letrd 11506	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
80	65, 62, 67, 79	lesubd 11960	. . . . . . . . . 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 11783	. . . . . . . . . 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 11972	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ≤ (𝐵 − 𝑘))
85	60, 61, 66, 80, 82, 84	fprodle 16207	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
86	4	nn0cnd 12709	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87		fallfacval 16220	. . . . . . . . . . 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 2769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) = (𝐴 FallFac 𝐶))
90	38	nn0cnd 12709	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91		fallfacval 16220	. . . . . . . . . . 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 2769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘) = (𝐵 FallFac 𝐶))
94	85, 89, 93	3brtr3d 5142	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95		fallfacval4 16253	. . . . . . . . 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 12745	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
98	27	nn0ge0d 12710	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
99	68, 12, 43, 15, 45	letrd 11506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
100	97, 41, 9, 98, 99	elfzd 13687	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101		fallfacval4 16253	. . . . . . . . 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 5142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
104	36, 58, 59, 103	lediv1dd 13262	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)))
105	39	nncnd 12391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
106	55	nncnd 12391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℂ)
107	105, 106, 29, 57, 31	divdiv1d 12164	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
108	104, 107	breqtrd 5137	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
109	33, 108	eqbrtrd 5133	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
110	37	nn0zd 12758	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
111	110	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112		elfzle1 13700	. . . . . . . 8 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113	112	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
114	3	nn0red 12708	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
115	114	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116	111	zred 12843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
117	11, 115, 116, 15, 45	letrd 11506	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
118	97, 111, 9, 113, 117	elfzd 13687	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119		bcval2 14490	. . . . . 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 2769	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122	109, 121	breqtrd 5137	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
123	2, 122	eqbrtrd 5133	. 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 14491	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
128	124, 125, 126, 127	syl3anc 1399	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
129		bccl2 14508	. . . . . . 7 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130	129	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131	130	nnnn0d 12707	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ0)
132	131	nn0ge0d 12710	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133		0le0 12484	. . . . . 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 14491	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
139	135, 136, 137, 138	syl3anc 1399	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
140	139	eqcomd 2769	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141	134, 140	breqtrd 5137	. . . 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 5133	. 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 1570   ∈ wcel 2147   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  ℝcr 11238  0cc0 11239  1c1 11240   · cmul 11244   ≤ cle 11383   − cmin 11580   / cdiv 12013  ℕcn 12375  ℕ₀cn0 12646  ℤcz 12733  ...cfz 13679  !cfa 14458  Ccbc 14487  ∏cprod 16116   FallFac cfallfac 16215

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-ico 13522  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-fac 14459  df-bc 14488  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-prod 16117  df-fallfac 16218

This theorem is used by:  aks6d1c7lem1  43496