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Theorem bcxmas 11368
 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 11367 . . . . 5 (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2 oveq2 5826 . . . . . 6 (𝑚 = 0 → (0...𝑚) = (0...0))
32sumeq1d 11245 . . . . 5 (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
41, 3eqeq12d 2172 . . . 4 (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
54imbi2d 229 . . 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 11367 . . . . 5 (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
7 oveq2 5826 . . . . . 6 (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
87sumeq1d 11245 . . . . 5 (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
96, 8eqeq12d 2172 . . . 4 (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
109imbi2d 229 . . 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 11367 . . . . 5 (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
12 oveq2 5826 . . . . . 6 (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
1312sumeq1d 11245 . . . . 5 (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
1411, 13eqeq12d 2172 . . . 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 229 . . 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 11367 . . . . 5 (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
17 oveq2 5826 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
1817sumeq1d 11245 . . . . 5 (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
1916, 18eqeq12d 2172 . . . 4 (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
2019imbi2d 229 . . 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 9088 . . . . 5 0 ∈ ℕ0
22 nn0addcl 9108 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
23 bcn0 10611 . . . . . 6 ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
2422, 23syl 14 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
2521, 24mpan2 422 . . . 4 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
26 0z 9161 . . . . 5 0 ∈ ℤ
27 1nn0 9089 . . . . . . 7 1 ∈ ℕ0
2825, 27eqeltrdi 2248 . . . . . 6 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
2928nn0cnd 9128 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
30 bcxmaslem1 11367 . . . . . 6 (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3130fsum1 11291 . . . . 5 ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3226, 29, 31sylancr 411 . . . 4 (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
33 peano2nn0 9113 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
34 nn0addcl 9108 . . . . . 6 (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
3533, 21, 34sylancl 410 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
36 bcn0 10611 . . . . 5 (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3735, 36syl 14 . . . 4 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3825, 32, 373eqtr4rd 2201 . . 3 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
39 simpr 109 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40 elnn0uz 9459 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ (ℤ‘0))
4139, 40sylib 121 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ‘0))
42 simpl 108 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
43 elfznn0 9998 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
44 nn0addcl 9108 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
4542, 43, 44syl2an 287 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
46 elfzelz 9910 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
4746adantl 275 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
48 bccl 10623 . . . . . . . . . . . 12 (((𝑁 + 𝑗) ∈ ℕ0𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
4945, 47, 48syl2anc 409 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
5049nn0cnd 9128 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
51 bcxmaslem1 11367 . . . . . . . . . 10 (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
5241, 50, 51fsump1 11299 . . . . . . . . 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 9083 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
5453adantr 274 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
55 nn0cn 9083 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
5655adantl 275 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
57 1cnd 7877 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 1 ∈ ℂ)
58 add32r 8018 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
5954, 56, 57, 58syl3anc 1220 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
6059oveq1d 5833 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
6160oveq2d 5834 . . . . . . . . 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 2190 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6362adantr 274 . . . . . . 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 5825 . . . . . . . 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 275 . . . . . . 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 7808 . . . . . . . . . . . . 13 1 ∈ ℂ
67 pncan 8064 . . . . . . . . . . . . 13 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
6856, 66, 67sylancl 410 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
6968oveq2d 5834 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
7069oveq2d 5834 . . . . . . . . . 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 9108 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
7233, 71sylan 281 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
73 nn0p1nn 9112 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
7473adantl 275 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
7574nnzd 9268 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
76 bcpasc 10622 . . . . . . . . . . 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 409 . . . . . . . . . 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 2192 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
79 nn0p1nn 9112 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
80 nnnn0addcl 9103 . . . . . . . . . . . . . 14 (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8179, 80sylan 281 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8281nnnn0d 9126 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
83 bccl 10623 . . . . . . . . . . . 12 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8482, 75, 83syl2anc 409 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8584nn0cnd 9128 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
86 nn0z 9170 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
8786adantl 275 . . . . . . . . . . . . 13 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
88 bccl 10623 . . . . . . . . . . . . 13 ((((𝑁 + 1) + 𝑘) ∈ ℕ0𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8971, 87, 88syl2anc 409 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9033, 89sylan 281 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9190nn0cnd 9128 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
9285, 91addcomd 8009 . . . . . . . . 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 7993 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
9453, 93syl 14 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
9594adantr 274 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
9695, 56, 57addassd 7883 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
9796oveq1d 5833 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9878, 92, 973eqtr3d 2198 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9998adantr 274 . . . . . . 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 2197 . . . . . 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 114 . . . . 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 115 . . . 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 9261 . 2 (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
105104impcom 124 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 103   = wceq 1335   ∈ wcel 2128  ‘cfv 5167  (class class class)co 5818  ℂcc 7713  0cc0 7715  1c1 7716   + caddc 7718   − cmin 8029  ℕcn 8816  ℕ0cn0 9073  ℤcz 9150  ℤ≥cuz 9422  ...cfz 9894  Ccbc 10603  Σcsu 11232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-en 6679  df-dom 6680  df-fin 6681  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-fac 10582  df-bc 10604  df-ihash 10632  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-sumdc 11233 This theorem is referenced by:  arisum  11377
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