bcled - Mathbox for metakunt

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Theorem bcled 42281

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 14222	. . . 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 14201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
6	5	nncnd 12151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
7	4	nn0zd 12504	. . . . . . . . . . . 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 12592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 𝐶) ∈ ℤ)
11	9	zred 12587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
12	4	nn0red 12453	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13		0red 11125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14		elfzle2 13438	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
15	14	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐴)
16	12	recnd 11150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
17	16	subid1d 11471	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
18	17	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
19	15, 18	breqtrd 5121	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
20	11, 12, 13, 19	lesubd 11731	. . . . . . . . . . 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 12491	. . . . . . . . . 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 14201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℕ)
25	24	nncnd 12151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℂ)
26		elfznn0 13530	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ₀)
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ₀)
28	27	faccld 14201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
29	28	nncnd 12151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
30	24	nnne0d 12185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ≠ 0)
31	28	nnne0d 12185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
32	6, 25, 29, 30, 31	divdiv1d 11938	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))))
33	32	eqcomd 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)))
34	5	nnred 12150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
35	24	nnred 12150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐴 − 𝐶)) ∈ ℝ)
36	34, 35, 30	redivcld 11959	. . . . . . 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 14201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
40	39	nnred 12150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
41	38	nn0zd 12504	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
42	41, 9	zsubcld 12592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
43	38	nn0red 12453	. . . . . . . . . . . . 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 11280	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
47	43	recnd 11150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
48	47	subid1d 11471	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
49	48	eqcomd 2739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
50	46, 49	breqtrd 5121	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
51	11, 43, 13, 50	lesubd 11731	. . . . . . . . . . . 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 12491	. . . . . . . . . . 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 14201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℕ)
56	55	nnred 12150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℝ)
57	55	nnne0d 12185	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ≠ 0)
58	40, 56, 57	redivcld 11959	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵 − 𝐶))) ∈ ℝ)
59	28	nnrpd 12942	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ⁺)
60		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ (0...𝐴))
61		fzfid 13890	. . . . . . . . . 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 13434	. . . . . . . . . . . . 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 12587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
66	62, 65	resubcld 11555	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ∈ ℝ)
67		0red 11125	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
68	27	nn0red 12453	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
69	68	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70		1red 11123	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
71	69, 70	resubcld 11555	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
72	62, 67	resubcld 11555	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73		elfzle2 13438	. . . . . . . . . . . . 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 11650	. . . . . . . . . . . . . 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 11747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
79	65, 71, 72, 74, 78	letrd 11280	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
80	65, 62, 67, 79	lesubd 11731	. . . . . . . . . 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 11555	. . . . . . . . . 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 11743	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 𝑘) ≤ (𝐵 − 𝑘))
85	60, 61, 66, 80, 82, 84	fprodle 15913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘))
86	4	nn0cnd 12454	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87		fallfacval 15926	. . . . . . . . . . 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 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴 − 𝑘) = (𝐴 FallFac 𝐶))
90	38	nn0cnd 12454	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91		fallfacval 15926	. . . . . . . . . . 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 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵 − 𝑘) = (𝐵 FallFac 𝐶))
94	85, 89, 93	3brtr3d 5126	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95		fallfacval4 15960	. . . . . . . . 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 12490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
98	27	nn0ge0d 12455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
99	68, 12, 43, 15, 45	letrd 11280	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
100	97, 41, 9, 98, 99	elfzd 13425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101		fallfacval4 15960	. . . . . . . . 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 5126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴 − 𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵 − 𝐶))))
104	36, 58, 59, 103	lediv1dd 13002	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)))
105	39	nncnd 12151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
106	55	nncnd 12151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (!‘(𝐵 − 𝐶)) ∈ ℂ)
107	105, 106, 29, 57, 31	divdiv1d 11938	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵 − 𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
108	104, 107	breqtrd 5121	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴 − 𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
109	33, 108	eqbrtrd 5117	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))))
110	37	nn0zd 12504	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
111	110	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112		elfzle1 13437	. . . . . . . 8 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113	112	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
114	3	nn0red 12453	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116	111	zred 12587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
117	11, 115, 116, 15, 45	letrd 11280	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ≤ 𝐵)
118	97, 111, 9, 113, 117	elfzd 13425	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119		bcval2 14222	. . . . . 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 2739	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵 − 𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122	109, 121	breqtrd 5121	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴 − 𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
123	2, 122	eqbrtrd 5117	. 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 14223	. . . 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 14240	. . . . . . 7 ⊢ (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130	129	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131	130	nnnn0d 12452	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ₀)
132	131	nn0ge0d 12455	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133		0le0 12236	. . . . . 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 14223	. . . . . . 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 2739	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141	134, 140	breqtrd 5121	. . . 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 5117	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
144	123, 143	pm2.61dan 812	1 ⊢ (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 class class class wbr 5095 ‘cfv 6489 (class class class)co 7355 ℂcc 11014 ℝcr 11015 0cc0 11016 1c1 11017 · cmul 11021 ≤ cle 11157 − cmin 11354 / cdiv 11784 ℕcn 12135 ℕ₀cn0 12391 ℤcz 12478 ...cfz 13417 !cfa 14190 Ccbc 14219 ∏cprod 15820 FallFac cfallfac 15921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9541 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-pre-sup 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-sup 9336 df-oi 9406 df-card 9842 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-div 11785 df-nn 12136 df-2 12198 df-3 12199 df-n0 12392 df-z 12479 df-uz 12743 df-rp 12901 df-ico 13261 df-fz 13418 df-fzo 13565 df-seq 13919 df-exp 13979 df-fac 14191 df-bc 14220 df-hash 14248 df-cj 15016 df-re 15017 df-im 15018 df-sqrt 15152 df-abs 15153 df-clim 15405 df-prod 15821 df-fallfac 15924

This theorem is referenced by: aks6d1c7lem1 42283

W3C validator