Theorem bcled 42631

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 14258	. . . 4 ⊢ (𝐶 ∈ (0...𝐴) → (𝐴C𝐶) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
2	1	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
3		bcled.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
4	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
5	4	faccld 14237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
6	5	nncnd 12181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
7	4	nn0zd 12540	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
8		bcled.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℤ)
9	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
10	7, 9	zsubcld 12629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℤ)
11	9	zred 12624	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
12	4	nn0red 12490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13		0red 11138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14		elfzle2 13473	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
15	14	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐴)
16	12	recnd 11164	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
17	16	subid1d 11485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
18	17	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
19	15, 18	breqtrd 5112	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
20	11, 12, 13, 19	lesubd 11745	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐴 − 𝐶))
21	10, 20	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐴 − 𝐶)))
22		elnn0z 12528	. . . . . . . . . 10 ⊢ ((𝐴 − 𝐶) ∈ ℕ0 ↔ ((𝐴 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐴 − 𝐶)))
23	21, 22	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℕ0)
24	23	faccld 14237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℕ)
25	24	nncnd 12181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℂ)
26		elfznn0 13565	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ0)
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ0)
28	27	faccld 14237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
29	28	nncnd 12181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
30	24	nnne0d 12218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ≠ 0)
31	28	nnne0d 12218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
32	6, 25, 29, 30, 31	divdiv1d 11953	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
33	32	eqcomd 2743	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)))
34	5	nnred 12180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
35	24	nnred 12180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℝ)
36	34, 35, 30	redivcld 11974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ∈ ℝ)
37		bcled.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℕ0)
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℕ0)
39	38	faccld 14237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
40	39	nnred 12180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
41	38	nn0zd 12540	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
42	41, 9	zsubcld 12629	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
43	38	nn0red 12490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
44		bcled.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
45	44	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ≤ 𝐵)
46	11, 12, 43, 15, 45	letrd 11294	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
47	43	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
48	47	subid1d 11485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
49	48	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
50	46, 49	breqtrd 5112	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
51	11, 43, 13, 50	lesubd 11745	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵 − 𝐶))
52	42, 51	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((𝐵 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐵 − 𝐶)))
53		elnn0z 12528	. . . . . . . . . . 11 ⊢ ((𝐵 − 𝐶) ∈ ℕ0 ↔ ((𝐵 − 𝐶) ∈ ℤ ∧ 0 ≤ (𝐵 − 𝐶)))
54	52, 53	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℕ0)
55	54	faccld 14237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℕ)
56	55	nnred 12180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℝ)
57	55	nnne0d 12218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ≠ 0)
58	40, 56, 57	redivcld 11974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵 − 𝐶))) ∈ ℝ)
59	28	nnrpd 12975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ+)
60		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ (0...𝐴))
61		fzfid 13926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (0...(𝐶 − 1)) ∈ Fin)
62	12	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ∈ ℝ)
63		elfzelz 13469	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ∈ ℤ)
64	63	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℤ)
65	64	zred 12624	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
66	62, 65	resubcld 11569	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ∈ ℝ)
67		0red 11138	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
68	27	nn0red 12490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
69	68	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70		1red 11136	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
71	69, 70	resubcld 11569	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
72	62, 67	resubcld 11569	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73		elfzle2 13473	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ≤ (𝐶 − 1))
74	73	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐶 − 1))
75	15	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ≤ 𝐴)
76		0le1 11664	. . . . . . . . . . . . . 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 11761	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
79	65, 71, 72, 74, 78	letrd 11294	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
80	65, 62, 67, 79	lesubd 11745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ (𝐴 − 𝑘))
81	43	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐵 ∈ ℝ)
82	81, 65	resubcld 11569	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐵 − 𝑘) ∈ ℝ)
83	44	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ≤ 𝐵)
84	62, 81, 65, 83	lesub1dd 11757	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ≤ (𝐵 − 𝑘))
85	60, 61, 66, 80, 82, 84	fprodle 15952	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
86	4	nn0cnd 12491	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87		fallfacval 15965	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘))
88	86, 27, 87	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘))
89	88	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) = (𝐴 FallFac 𝐶))
90	38	nn0cnd 12491	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91		fallfacval 15965	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
92	90, 27, 91	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
93	92	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘) = (𝐵 FallFac 𝐶))
94	85, 89, 93	3brtr3d 5117	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95		fallfacval4 15999	. . . . . . . . 9 ⊢ (𝐶 ∈ (0...𝐴) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴 − 𝐶))))
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴 − 𝐶))))
97		0zd 12527	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
98	27	nn0ge0d 12492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
99	68, 12, 43, 15, 45	letrd 11294	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
100	97, 41, 9, 98, 99	elfzd 13460	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101		fallfacval4 15999	. . . . . . . . 9 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
102	100, 101	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
103	94, 96, 102	3brtr3d 5117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
104	36, 58, 59, 103	lediv1dd 13035	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)))
105	39	nncnd 12181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
106	55	nncnd 12181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℂ)
107	105, 106, 29, 57, 31	divdiv1d 11953	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
108	104, 107	breqtrd 5112	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
109	33, 108	eqbrtrd 5108	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
110	37	nn0zd 12540	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
111	110	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112		elfzle1 13472	. . . . . . . 8 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113	112	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
114	3	nn0red 12490	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116	111	zred 12624	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
117	11, 115, 116, 15, 45	letrd 11294	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
118	97, 111, 9, 113, 117	elfzd 13460	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119		bcval2 14258	. . . . . 6 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
120	118, 119	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
121	120	eqcomd 2743	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122	109, 121	breqtrd 5112	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
123	2, 122	eqbrtrd 5108	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
124	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
125	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
126		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → ¬ 𝐶 ∈ (0...𝐴))
127		bcval3 14259	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
128	124, 125, 126, 127	syl3anc 1374	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
129		bccl2 14276	. . . . . . 7 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130	129	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131	130	nnnn0d 12489	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ0)
132	131	nn0ge0d 12492	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133		0le0 12273	. . . . . 6 ⊢ 0 ≤ 0
134	133	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ 0)
135	37	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 𝐵 ∈ ℕ0)
136	125	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 𝐶 ∈ ℤ)
137		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → ¬ 𝐶 ∈ (0...𝐵))
138		bcval3 14259	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
139	135, 136, 137, 138	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
140	139	eqcomd 2743	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141	134, 140	breqtrd 5112	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
142	132, 141	pm2.61dan 813	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵C𝐶))
143	128, 142	eqbrtrd 5108	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
144	123, 143	pm2.61dan 813	1 ⊢ (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   · cmul 11034   ≤ cle 11171   − cmin 11368   / cdiv 11798  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ...cfz 13452  !cfa 14226  Ccbc 14255  ∏cprod 15859   FallFac cfallfac 15960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-ico 13295  df-fz 13453  df-fzo 13600  df-seq 13955  df-exp 14015  df-fac 14227  df-bc 14256  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-prod 15860  df-fallfac 15963

This theorem is referenced by:  aks6d1c7lem1  42633