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Theorem bcxmas 10883
Description: Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
Assertion
Ref Expression
bcxmas ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
Distinct variable groups:   𝑗,𝑀   𝑗,𝑁

Proof of Theorem bcxmas
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcxmaslem1 10882 . . . . 5 (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2 oveq2 5660 . . . . . 6 (𝑚 = 0 → (0...𝑚) = (0...0))
32sumeq1d 10755 . . . . 5 (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
41, 3eqeq12d 2102 . . . 4 (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
54imbi2d 228 . . 3 (𝑚 = 0 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))))
6 bcxmaslem1 10882 . . . . 5 (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
7 oveq2 5660 . . . . . 6 (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
87sumeq1d 10755 . . . . 5 (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
96, 8eqeq12d 2102 . . . 4 (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
109imbi2d 228 . . 3 (𝑚 = 𝑘 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))))
11 bcxmaslem1 10882 . . . . 5 (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
12 oveq2 5660 . . . . . 6 (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
1312sumeq1d 10755 . . . . 5 (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
1411, 13eqeq12d 2102 . . . 4 (𝑚 = (𝑘 + 1) → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗)))
1514imbi2d 228 . . 3 (𝑚 = (𝑘 + 1) → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))))
16 bcxmaslem1 10882 . . . . 5 (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
17 oveq2 5660 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
1817sumeq1d 10755 . . . . 5 (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
1916, 18eqeq12d 2102 . . . 4 (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
2019imbi2d 228 . . 3 (𝑚 = 𝑀 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))))
21 0nn0 8688 . . . . 5 0 ∈ ℕ0
22 nn0addcl 8708 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
23 bcn0 10163 . . . . . 6 ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
2422, 23syl 14 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
2521, 24mpan2 416 . . . 4 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
26 0z 8761 . . . . 5 0 ∈ ℤ
27 1nn0 8689 . . . . . . 7 1 ∈ ℕ0
2825, 27syl6eqel 2178 . . . . . 6 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
2928nn0cnd 8728 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
30 bcxmaslem1 10882 . . . . . 6 (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3130fsum1 10806 . . . . 5 ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3226, 29, 31sylancr 405 . . . 4 (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
33 peano2nn0 8713 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
34 nn0addcl 8708 . . . . . 6 (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
3533, 21, 34sylancl 404 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
36 bcn0 10163 . . . . 5 (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3735, 36syl 14 . . . 4 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3825, 32, 373eqtr4rd 2131 . . 3 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
39 simpr 108 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40 elnn0uz 9056 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ (ℤ‘0))
4139, 40sylib 120 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ‘0))
42 simpl 107 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
43 elfznn0 9528 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
44 nn0addcl 8708 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
4542, 43, 44syl2an 283 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
46 elfzelz 9440 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
4746adantl 271 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
48 bccl 10175 . . . . . . . . . . . 12 (((𝑁 + 𝑗) ∈ ℕ0𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
4945, 47, 48syl2anc 403 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
5049nn0cnd 8728 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
51 bcxmaslem1 10882 . . . . . . . . . 10 (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
5241, 50, 51fsump1 10814 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))))
53 nn0cn 8683 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
5453adantr 270 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
55 nn0cn 8683 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
5655adantl 271 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
57 1cnd 7504 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 1 ∈ ℂ)
58 add32r 7642 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
5954, 56, 57, 58syl3anc 1174 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
6059oveq1d 5667 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
6160oveq2d 5668 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6252, 61eqtrd 2120 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6362adantr 270 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
64 oveq1 5659 . . . . . . . 8 ((((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6564adantl 271 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
66 ax-1cn 7438 . . . . . . . . . . . . 13 1 ∈ ℂ
67 pncan 7688 . . . . . . . . . . . . 13 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
6856, 66, 67sylancl 404 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
6968oveq2d 5668 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
7069oveq2d 5668 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)))
71 nn0addcl 8708 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
7233, 71sylan 277 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
73 nn0p1nn 8712 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
7473adantl 271 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
7574nnzd 8867 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
76 bcpasc 10174 . . . . . . . . . . 11 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
7772, 75, 76syl2anc 403 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
7870, 77eqtr3d 2122 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
79 nn0p1nn 8712 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
80 nnnn0addcl 8703 . . . . . . . . . . . . . 14 (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8179, 80sylan 277 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8281nnnn0d 8726 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
83 bccl 10175 . . . . . . . . . . . 12 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8482, 75, 83syl2anc 403 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8584nn0cnd 8728 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
86 nn0z 8770 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
8786adantl 271 . . . . . . . . . . . . 13 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
88 bccl 10175 . . . . . . . . . . . . 13 ((((𝑁 + 1) + 𝑘) ∈ ℕ0𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8971, 87, 88syl2anc 403 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9033, 89sylan 277 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9190nn0cnd 8728 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
9285, 91addcomd 7633 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
93 peano2cn 7617 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
9453, 93syl 14 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
9594adantr 270 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
9695, 56, 57addassd 7510 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
9796oveq1d 5667 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9878, 92, 973eqtr3d 2128 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9998adantr 270 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
10063, 65, 993eqtr2rd 2127 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
101100ex 113 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗)))
102101expcom 114 . . . 4 (𝑘 ∈ ℕ0 → (𝑁 ∈ ℕ0 → ((((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))))
103102a2d 26 . . 3 (𝑘 ∈ ℕ0 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))))
1045, 10, 15, 20, 38, 103nn0ind 8860 . 2 (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
105104impcom 123 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cfv 5015  (class class class)co 5652  cc 7348  0cc0 7350  1c1 7351   + caddc 7353  cmin 7653  cn 8422  0cn0 8673  cz 8750  cuz 9019  ...cfz 9424  Ccbc 10155  Σcsu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-fac 10134  df-bc 10156  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by:  arisum  10892
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