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Theorem bcled 42179
Description: Inequality for binomial coefficients. (Contributed by metakunt, 12-May-2025.)
Hypotheses
Ref Expression
bcled.1 (𝜑𝐴 ∈ ℕ0)
bcled.2 (𝜑𝐵 ∈ ℕ0)
bcled.3 (𝜑𝐶 ∈ ℤ)
bcled.4 (𝜑𝐴𝐵)
Assertion
Ref Expression
bcled (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))

Proof of Theorem bcled
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 bcval2 14344 . . . 4 (𝐶 ∈ (0...𝐴) → (𝐴C𝐶) = ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))))
21adantl 481 . . 3 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))))
3 bcled.1 . . . . . . . . . 10 (𝜑𝐴 ∈ ℕ0)
43adantr 480 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
54faccld 14323 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
65nncnd 12282 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
74nn0zd 12639 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
8 bcled.3 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℤ)
98adantr 480 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
107, 9zsubcld 12727 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴𝐶) ∈ ℤ)
119zred 12722 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
124nn0red 12588 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13 0red 11264 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14 elfzle2 13568 . . . . . . . . . . . . . 14 (𝐶 ∈ (0...𝐴) → 𝐶𝐴)
1514adantl 481 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐴)
1612recnd 11289 . . . . . . . . . . . . . . 15 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
1716subid1d 11609 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
1817eqcomd 2743 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
1915, 18breqtrd 5169 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
2011, 12, 13, 19lesubd 11867 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐴𝐶))
2110, 20jca 511 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → ((𝐴𝐶) ∈ ℤ ∧ 0 ≤ (𝐴𝐶)))
22 elnn0z 12626 . . . . . . . . . 10 ((𝐴𝐶) ∈ ℕ0 ↔ ((𝐴𝐶) ∈ ℤ ∧ 0 ≤ (𝐴𝐶)))
2321, 22sylibr 234 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴𝐶) ∈ ℕ0)
2423faccld 14323 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ∈ ℕ)
2524nncnd 12282 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ∈ ℂ)
26 elfznn0 13660 . . . . . . . . . 10 (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ0)
2726adantl 481 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ0)
2827faccld 14323 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
2928nncnd 12282 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
3024nnne0d 12316 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ≠ 0)
3128nnne0d 12316 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
326, 25, 29, 30, 31divdiv1d 12074 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))))
3332eqcomd 2743 . . . . 5 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)))
345nnred 12281 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
3524nnred 12281 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ∈ ℝ)
3634, 35, 30redivcld 12095 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴𝐶))) ∈ ℝ)
37 bcled.2 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℕ0)
3837adantr 480 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℕ0)
3938faccld 14323 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
4039nnred 12281 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
4138nn0zd 12639 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
4241, 9zsubcld 12727 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℤ)
4338nn0red 12588 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
44 bcled.4 . . . . . . . . . . . . . . . 16 (𝜑𝐴𝐵)
4544adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴𝐵)
4611, 12, 43, 15, 45letrd 11418 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐵)
4743recnd 11289 . . . . . . . . . . . . . . . 16 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
4847subid1d 11609 . . . . . . . . . . . . . . 15 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
4948eqcomd 2743 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
5046, 49breqtrd 5169 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
5111, 43, 13, 50lesubd 11867 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵𝐶))
5242, 51jca 511 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → ((𝐵𝐶) ∈ ℤ ∧ 0 ≤ (𝐵𝐶)))
53 elnn0z 12626 . . . . . . . . . . 11 ((𝐵𝐶) ∈ ℕ0 ↔ ((𝐵𝐶) ∈ ℤ ∧ 0 ≤ (𝐵𝐶)))
5452, 53sylibr 234 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℕ0)
5554faccld 14323 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ∈ ℕ)
5655nnred 12281 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ∈ ℝ)
5755nnne0d 12316 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ≠ 0)
5840, 56, 57redivcld 12095 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵𝐶))) ∈ ℝ)
5928nnrpd 13075 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ+)
60 nfv 1914 . . . . . . . . . 10 𝑘(𝜑𝐶 ∈ (0...𝐴))
61 fzfid 14014 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → (0...(𝐶 − 1)) ∈ Fin)
6212adantr 480 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ∈ ℝ)
63 elfzelz 13564 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ∈ ℤ)
6463adantl 481 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℤ)
6564zred 12722 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
6662, 65resubcld 11691 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴𝑘) ∈ ℝ)
67 0red 11264 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
6827nn0red 12588 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
6968adantr 480 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70 1red 11262 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
7169, 70resubcld 11691 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
7262, 67resubcld 11691 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73 elfzle2 13568 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ≤ (𝐶 − 1))
7473adantl 481 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐶 − 1))
7515adantr 480 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶𝐴)
76 0le1 11786 . . . . . . . . . . . . . 14 0 ≤ 1
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ 1)
7869, 67, 62, 70, 75, 77le2subd 11883 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
7965, 71, 72, 74, 78letrd 11418 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
8065, 62, 67, 79lesubd 11867 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ (𝐴𝑘))
8143adantr 480 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐵 ∈ ℝ)
8281, 65resubcld 11691 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐵𝑘) ∈ ℝ)
8344ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴𝐵)
8462, 81, 65, 83lesub1dd 11879 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴𝑘) ≤ (𝐵𝑘))
8560, 61, 66, 80, 82, 84fprodle 16032 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵𝑘))
864nn0cnd 12589 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87 fallfacval 16045 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴𝑘))
8886, 27, 87syl2anc 584 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴𝑘))
8988eqcomd 2743 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴𝑘) = (𝐴 FallFac 𝐶))
9038nn0cnd 12589 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91 fallfacval 16045 . . . . . . . . . . 11 ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵𝑘))
9290, 27, 91syl2anc 584 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵𝑘))
9392eqcomd 2743 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵𝑘) = (𝐵 FallFac 𝐶))
9485, 89, 933brtr3d 5174 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95 fallfacval4 16079 . . . . . . . . 9 (𝐶 ∈ (0...𝐴) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴𝐶))))
9695adantl 481 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴𝐶))))
97 0zd 12625 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
9827nn0ge0d 12590 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
9968, 12, 43, 15, 45letrd 11418 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐵)
10097, 41, 9, 98, 99elfzd 13555 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101 fallfacval4 16079 . . . . . . . . 9 (𝐶 ∈ (0...𝐵) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵𝐶))))
102100, 101syl 17 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵𝐶))))
10394, 96, 1023brtr3d 5174 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵𝐶))))
10436, 58, 59, 103lediv1dd 13135 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵𝐶))) / (!‘𝐶)))
10539nncnd 12282 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
10655nncnd 12282 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ∈ ℂ)
107105, 106, 29, 57, 31divdiv1d 12074 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
108104, 107breqtrd 5169 . . . . 5 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
10933, 108eqbrtrd 5165 . . . 4 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
11037nn0zd 12639 . . . . . . . 8 (𝜑𝐵 ∈ ℤ)
111110adantr 480 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112 elfzle1 13567 . . . . . . . 8 (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113112adantl 481 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
1143nn0red 12588 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
115114adantr 480 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116111zred 12722 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
11711, 115, 116, 15, 45letrd 11418 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐵)
11897, 111, 9, 113, 117elfzd 13555 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119 bcval2 14344 . . . . . 6 (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
120118, 119syl 17 . . . . 5 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
121120eqcomd 2743 . . . 4 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122109, 121breqtrd 5169 . . 3 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
1232, 122eqbrtrd 5165 . 2 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
1243adantr 480 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
1258adantr 480 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
126 simpr 484 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → ¬ 𝐶 ∈ (0...𝐴))
127 bcval3 14345 . . . 4 ((𝐴 ∈ ℕ0𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
128124, 125, 126, 127syl3anc 1373 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
129 bccl2 14362 . . . . . . 7 (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130129adantl 481 . . . . . 6 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131130nnnn0d 12587 . . . . 5 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ0)
132131nn0ge0d 12590 . . . 4 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133 0le0 12367 . . . . . 6 0 ≤ 0
134133a1i 11 . . . . 5 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ 0)
13537ad2antrr 726 . . . . . . 7 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 𝐵 ∈ ℕ0)
136125adantr 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 14345 . . . . . . 7 ((𝐵 ∈ ℕ0𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
139135, 136, 137, 138syl3anc 1373 . . . . . 6 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
140139eqcomd 2743 . . . . 5 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141134, 140breqtrd 5169 . . . 4 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
142132, 141pm2.61dan 813 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵C𝐶))
143128, 142eqbrtrd 5165 . 2 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) ≤ (𝐵C𝐶))
144123, 143pm2.61dan 813 1 (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   · cmul 11160  cle 11296  cmin 11492   / cdiv 11920  cn 12266  0cn0 12526  cz 12613  ...cfz 13547  !cfa 14312  Ccbc 14341  cprod 15939   FallFac cfallfac 16040
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-prod 15940  df-fallfac 16043
This theorem is referenced by:  aks6d1c7lem1  42181
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