Theorem bcled 42139

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 14246	. . . 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 14225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
6	5	nncnd 12178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
7	4	nn0zd 12531	. . . . . . . . . . . 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 12619	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℤ)
11	9	zred 12614	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
12	4	nn0red 12480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13		0red 11153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14		elfzle2 13465	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
15	14	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐴)
16	12	recnd 11178	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
17	16	subid1d 11498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
18	17	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
19	15, 18	breqtrd 5128	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
20	11, 12, 13, 19	lesubd 11758	. . . . . . . . . . 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 12518	. . . . . . . . . 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 14225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℕ)
25	24	nncnd 12178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℂ)
26		elfznn0 13557	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ0)
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ0)
28	27	faccld 14225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
29	28	nncnd 12178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
30	24	nnne0d 12212	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ≠ 0)
31	28	nnne0d 12212	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
32	6, 25, 29, 30, 31	divdiv1d 11965	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
33	32	eqcomd 2735	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)))
34	5	nnred 12177	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
35	24	nnred 12177	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℝ)
36	34, 35, 30	redivcld 11986	. . . . . . 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 14225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
40	39	nnred 12177	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
41	38	nn0zd 12531	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
42	41, 9	zsubcld 12619	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
43	38	nn0red 12480	. . . . . . . . . . . . 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 11307	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
47	43	recnd 11178	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
48	47	subid1d 11498	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
49	48	eqcomd 2735	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
50	46, 49	breqtrd 5128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
51	11, 43, 13, 50	lesubd 11758	. . . . . . . . . . . 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 12518	. . . . . . . . . . 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 14225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℕ)
56	55	nnred 12177	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℝ)
57	55	nnne0d 12212	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ≠ 0)
58	40, 56, 57	redivcld 11986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵 − 𝐶))) ∈ ℝ)
59	28	nnrpd 12969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ+)
60		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ (0...𝐴))
61		fzfid 13914	. . . . . . . . . 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 13461	. . . . . . . . . . . . 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 12614	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
66	62, 65	resubcld 11582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ∈ ℝ)
67		0red 11153	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
68	27	nn0red 12480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
69	68	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70		1red 11151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
71	69, 70	resubcld 11582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
72	62, 67	resubcld 11582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73		elfzle2 13465	. . . . . . . . . . . . 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 11677	. . . . . . . . . . . . . 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 11774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
79	65, 71, 72, 74, 78	letrd 11307	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
80	65, 62, 67, 79	lesubd 11758	. . . . . . . . . 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 11582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐵 − 𝑘) ∈ ℝ)
83	44	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ≤ 𝐵)
84	62, 81, 65, 83	lesub1dd 11770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ≤ (𝐵 − 𝑘))
85	60, 61, 66, 80, 82, 84	fprodle 15938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
86	4	nn0cnd 12481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87		fallfacval 15951	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘))
88	86, 27, 87	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘))
89	88	eqcomd 2735	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) = (𝐴 FallFac 𝐶))
90	38	nn0cnd 12481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91		fallfacval 15951	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
92	90, 27, 91	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
93	92	eqcomd 2735	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘) = (𝐵 FallFac 𝐶))
94	85, 89, 93	3brtr3d 5133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95		fallfacval4 15985	. . . . . . . . 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 12517	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
98	27	nn0ge0d 12482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
99	68, 12, 43, 15, 45	letrd 11307	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
100	97, 41, 9, 98, 99	elfzd 13452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101		fallfacval4 15985	. . . . . . . . 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 5133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
104	36, 58, 59, 103	lediv1dd 13029	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)))
105	39	nncnd 12178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
106	55	nncnd 12178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℂ)
107	105, 106, 29, 57, 31	divdiv1d 11965	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
108	104, 107	breqtrd 5128	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
109	33, 108	eqbrtrd 5124	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
110	37	nn0zd 12531	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
111	110	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112		elfzle1 13464	. . . . . . . 8 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113	112	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
114	3	nn0red 12480	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116	111	zred 12614	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
117	11, 115, 116, 15, 45	letrd 11307	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
118	97, 111, 9, 113, 117	elfzd 13452	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119		bcval2 14246	. . . . . 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 2735	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122	109, 121	breqtrd 5128	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
123	2, 122	eqbrtrd 5124	. 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 14247	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
128	124, 125, 126, 127	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
129		bccl2 14264	. . . . . . 7 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130	129	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131	130	nnnn0d 12479	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ0)
132	131	nn0ge0d 12482	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133		0le0 12263	. . . . . 6 ⊢ 0 ≤ 0
134	133	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ 0)
135	37	ad2antrr 726	. . . . . . 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 14247	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
139	135, 136, 137, 138	syl3anc 1373	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
140	139	eqcomd 2735	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141	134, 140	breqtrd 5128	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
142	132, 141	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵C𝐶))
143	128, 142	eqbrtrd 5124	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
144	123, 143	pm2.61dan 812	1 ⊢ (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   · cmul 11049   ≤ cle 11185   − cmin 11381   / cdiv 11811  ℕcn 12162  ℕ₀cn0 12418  ℤcz 12505  ...cfz 13444  !cfa 14214  Ccbc 14243  ∏cprod 15845   FallFac cfallfac 15946

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-ico 13288  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-fac 14215  df-bc 14244  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-prod 15846  df-fallfac 15949

This theorem is referenced by:  aks6d1c7lem1  42141