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Theorem bcled 42159
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 14340 . . . 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 14319 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℕ)
65nncnd 12279 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℂ)
74nn0zd 12636 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
8 bcled.3 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℤ)
98adantr 480 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
107, 9zsubcld 12724 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴𝐶) ∈ ℤ)
119zred 12719 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
124nn0red 12585 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
13 0red 11261 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ∈ ℝ)
14 elfzle2 13564 . . . . . . . . . . . . . 14 (𝐶 ∈ (0...𝐴) → 𝐶𝐴)
1514adantl 481 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐴)
1612recnd 11286 . . . . . . . . . . . . . . 15 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
1716subid1d 11606 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
1817eqcomd 2740 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 = (𝐴 − 0))
1915, 18breqtrd 5173 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐴 − 0))
2011, 12, 13, 19lesubd 11864 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐴𝐶))
2110, 20jca 511 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → ((𝐴𝐶) ∈ ℤ ∧ 0 ≤ (𝐴𝐶)))
22 elnn0z 12623 . . . . . . . . . 10 ((𝐴𝐶) ∈ ℕ0 ↔ ((𝐴𝐶) ∈ ℤ ∧ 0 ≤ (𝐴𝐶)))
2321, 22sylibr 234 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴𝐶) ∈ ℕ0)
2423faccld 14319 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ∈ ℕ)
2524nncnd 12279 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ∈ ℂ)
26 elfznn0 13656 . . . . . . . . . 10 (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℕ0)
2726adantl 481 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℕ0)
2827faccld 14319 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℕ)
2928nncnd 12279 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℂ)
3024nnne0d 12313 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ≠ 0)
3128nnne0d 12313 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ≠ 0)
326, 25, 29, 30, 31divdiv1d 12071 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)) = ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))))
3332eqcomd 2740 . . . . 5 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))) = (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)))
345nnred 12278 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐴) ∈ ℝ)
3524nnred 12278 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐴𝐶)) ∈ ℝ)
3634, 35, 30redivcld 12092 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴𝐶))) ∈ ℝ)
37 bcled.2 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℕ0)
3837adantr 480 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℕ0)
3938faccld 14319 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℕ)
4039nnred 12278 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℝ)
4138nn0zd 12636 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
4241, 9zsubcld 12724 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℤ)
4338nn0red 12585 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
44 bcled.4 . . . . . . . . . . . . . . . 16 (𝜑𝐴𝐵)
4544adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴𝐵)
4611, 12, 43, 15, 45letrd 11415 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐵)
4743recnd 11286 . . . . . . . . . . . . . . . 16 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
4847subid1d 11606 . . . . . . . . . . . . . . 15 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵 − 0) = 𝐵)
4948eqcomd 2740 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 = (𝐵 − 0))
5046, 49breqtrd 5173 . . . . . . . . . . . . 13 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ≤ (𝐵 − 0))
5111, 43, 13, 50lesubd 11864 . . . . . . . . . . . 12 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵𝐶))
5242, 51jca 511 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → ((𝐵𝐶) ∈ ℤ ∧ 0 ≤ (𝐵𝐶)))
53 elnn0z 12623 . . . . . . . . . . 11 ((𝐵𝐶) ∈ ℕ0 ↔ ((𝐵𝐶) ∈ ℤ ∧ 0 ≤ (𝐵𝐶)))
5452, 53sylibr 234 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℕ0)
5554faccld 14319 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ∈ ℕ)
5655nnred 12278 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ∈ ℝ)
5755nnne0d 12313 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ≠ 0)
5840, 56, 57redivcld 12092 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / (!‘(𝐵𝐶))) ∈ ℝ)
5928nnrpd 13072 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐶) ∈ ℝ+)
60 nfv 1911 . . . . . . . . . 10 𝑘(𝜑𝐶 ∈ (0...𝐴))
61 fzfid 14010 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → (0...(𝐶 − 1)) ∈ Fin)
6212adantr 480 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐴 ∈ ℝ)
63 elfzelz 13560 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝐶 − 1)) → 𝑘 ∈ ℤ)
6463adantl 481 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℤ)
6564zred 12719 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ∈ ℝ)
6662, 65resubcld 11688 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴𝑘) ∈ ℝ)
67 0red 11261 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ∈ ℝ)
6827nn0red 12585 . . . . . . . . . . . . . 14 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
6968adantr 480 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐶 ∈ ℝ)
70 1red 11259 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 1 ∈ ℝ)
7169, 70resubcld 11688 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ∈ ℝ)
7262, 67resubcld 11688 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴 − 0) ∈ ℝ)
73 elfzle2 13564 . . . . . . . . . . . . 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 11783 . . . . . . . . . . . . . 14 0 ≤ 1
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ 1)
7869, 67, 62, 70, 75, 77le2subd 11880 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐶 − 1) ≤ (𝐴 − 0))
7965, 71, 72, 74, 78letrd 11415 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝑘 ≤ (𝐴 − 0))
8065, 62, 67, 79lesubd 11864 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 0 ≤ (𝐴𝑘))
8143adantr 480 . . . . . . . . . . 11 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → 𝐵 ∈ ℝ)
8281, 65resubcld 11688 . . . . . . . . . 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 11876 . . . . . . . . . 10 (((𝜑𝐶 ∈ (0...𝐴)) ∧ 𝑘 ∈ (0...(𝐶 − 1))) → (𝐴𝑘) ≤ (𝐵𝑘))
8560, 61, 66, 80, 82, 84fprodle 16028 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴𝑘) ≤ ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵𝑘))
864nn0cnd 12586 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
87 fallfacval 16041 . . . . . . . . . . 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 2740 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐴𝑘) = (𝐴 FallFac 𝐶))
9038nn0cnd 12586 . . . . . . . . . . 11 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
91 fallfacval 16041 . . . . . . . . . . 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 2740 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → ∏𝑘 ∈ (0...(𝐶 − 1))(𝐵𝑘) = (𝐵 FallFac 𝐶))
9485, 89, 933brtr3d 5178 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) ≤ (𝐵 FallFac 𝐶))
95 fallfacval4 16075 . . . . . . . . 9 (𝐶 ∈ (0...𝐴) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴𝐶))))
9695adantl 481 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐴 FallFac 𝐶) = ((!‘𝐴) / (!‘(𝐴𝐶))))
97 0zd 12622 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
9827nn0ge0d 12587 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
9968, 12, 43, 15, 45letrd 11415 . . . . . . . . . 10 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐵)
10097, 41, 9, 98, 99elfzd 13551 . . . . . . . . 9 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
101 fallfacval4 16075 . . . . . . . . 9 (𝐶 ∈ (0...𝐵) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵𝐶))))
102100, 101syl 17 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵 FallFac 𝐶) = ((!‘𝐵) / (!‘(𝐵𝐶))))
10394, 96, 1023brtr3d 5178 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / (!‘(𝐴𝐶))) ≤ ((!‘𝐵) / (!‘(𝐵𝐶))))
10436, 58, 59, 103lediv1dd 13132 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)) ≤ (((!‘𝐵) / (!‘(𝐵𝐶))) / (!‘𝐶)))
10539nncnd 12279 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘𝐵) ∈ ℂ)
10655nncnd 12279 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → (!‘(𝐵𝐶)) ∈ ℂ)
107105, 106, 29, 57, 31divdiv1d 12071 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐵) / (!‘(𝐵𝐶))) / (!‘𝐶)) = ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
108104, 107breqtrd 5173 . . . . 5 ((𝜑𝐶 ∈ (0...𝐴)) → (((!‘𝐴) / (!‘(𝐴𝐶))) / (!‘𝐶)) ≤ ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
10933, 108eqbrtrd 5169 . . . 4 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))) ≤ ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
11037nn0zd 12636 . . . . . . . 8 (𝜑𝐵 ∈ ℤ)
111110adantr 480 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
112 elfzle1 13563 . . . . . . . 8 (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
113112adantl 481 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → 0 ≤ 𝐶)
1143nn0red 12585 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
115114adantr 480 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
116111zred 12719 . . . . . . . 8 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
11711, 115, 116, 15, 45letrd 11415 . . . . . . 7 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶𝐵)
11897, 111, 9, 113, 117elfzd 13551 . . . . . 6 ((𝜑𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐵))
119 bcval2 14340 . . . . . 6 (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
120118, 119syl 17 . . . . 5 ((𝜑𝐶 ∈ (0...𝐴)) → (𝐵C𝐶) = ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))))
121120eqcomd 2740 . . . 4 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐵) / ((!‘(𝐵𝐶)) · (!‘𝐶))) = (𝐵C𝐶))
122109, 121breqtrd 5173 . . 3 ((𝜑𝐶 ∈ (0...𝐴)) → ((!‘𝐴) / ((!‘(𝐴𝐶)) · (!‘𝐶))) ≤ (𝐵C𝐶))
1232, 122eqbrtrd 5169 . 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 14341 . . . 4 ((𝐴 ∈ ℕ0𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
128124, 125, 126, 127syl3anc 1370 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → (𝐴C𝐶) = 0)
129 bccl2 14358 . . . . . . 7 (𝐶 ∈ (0...𝐵) → (𝐵C𝐶) ∈ ℕ)
130129adantl 481 . . . . . 6 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ)
131130nnnn0d 12584 . . . . 5 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) ∈ ℕ0)
132131nn0ge0d 12587 . . . 4 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
133 0le0 12364 . . . . . 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 14341 . . . . . . 7 ((𝐵 ∈ ℕ0𝐶 ∈ ℤ ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
139135, 136, 137, 138syl3anc 1370 . . . . . 6 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → (𝐵C𝐶) = 0)
140139eqcomd 2740 . . . . 5 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 = (𝐵C𝐶))
141134, 140breqtrd 5173 . . . 4 (((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) ∧ ¬ 𝐶 ∈ (0...𝐵)) → 0 ≤ (𝐵C𝐶))
142132, 141pm2.61dan 813 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ (0...𝐴)) → 0 ≤ (𝐵C𝐶))
143128, 142eqbrtrd 5169 . 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 1536  wcel 2105   class class class wbr 5147  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   · cmul 11157  cle 11293  cmin 11489   / cdiv 11917  cn 12263  0cn0 12523  cz 12610  ...cfz 13543  !cfa 14308  Ccbc 14337  cprod 15935   FallFac cfallfac 16036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-prod 15936  df-fallfac 16039
This theorem is referenced by:  aks6d1c7lem1  42161
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