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Theorem bcxmas 14354
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 14353 . . . . 5 (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2 oveq2 6534 . . . . . 6 (𝑚 = 0 → (0...𝑚) = (0...0))
32sumeq1d 14227 . . . . 5 (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
41, 3eqeq12d 2624 . . . 4 (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
54imbi2d 328 . . 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 14353 . . . . 5 (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
7 oveq2 6534 . . . . . 6 (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
87sumeq1d 14227 . . . . 5 (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
96, 8eqeq12d 2624 . . . 4 (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
109imbi2d 328 . . 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 14353 . . . . 5 (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
12 oveq2 6534 . . . . . 6 (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
1312sumeq1d 14227 . . . . 5 (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
1411, 13eqeq12d 2624 . . . 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 328 . . 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 14353 . . . . 5 (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
17 oveq2 6534 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
1817sumeq1d 14227 . . . . 5 (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
1916, 18eqeq12d 2624 . . . 4 (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
2019imbi2d 328 . . 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 11156 . . . . 5 0 ∈ ℕ0
22 nn0addcl 11177 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
23 bcn0 12916 . . . . . 6 ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
2422, 23syl 17 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
2521, 24mpan2 702 . . . 4 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
26 0z 11223 . . . . 5 0 ∈ ℤ
27 1nn0 11157 . . . . . . 7 1 ∈ ℕ0
2825, 27syl6eqel 2695 . . . . . 6 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
2928nn0cnd 11202 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
30 bcxmaslem1 14353 . . . . . 6 (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3130fsum1 14268 . . . . 5 ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3226, 29, 31sylancr 693 . . . 4 (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
33 peano2nn0 11182 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
34 nn0addcl 11177 . . . . . 6 (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
3533, 21, 34sylancl 692 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
36 bcn0 12916 . . . . 5 (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3735, 36syl 17 . . . 4 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3825, 32, 373eqtr4rd 2654 . . 3 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
39 simpr 475 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40 elnn0uz 11559 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ (ℤ‘0))
4139, 40sylib 206 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ‘0))
42 simpl 471 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
43 elfznn0 12259 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
44 nn0addcl 11177 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
4542, 43, 44syl2an 492 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
46 elfzelz 12170 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
4746adantl 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
48 bccl 12928 . . . . . . . . . . . 12 (((𝑁 + 𝑗) ∈ ℕ0𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
4945, 47, 48syl2anc 690 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
5049nn0cnd 11202 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
51 bcxmaslem1 14353 . . . . . . . . . 10 (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
5241, 50, 51fsump1 14277 . . . . . . . . 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 11151 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
5453adantr 479 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
55 nn0cn 11151 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
5655adantl 480 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
57 1cnd 9912 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 1 ∈ ℂ)
58 add32r 10106 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
5954, 56, 57, 58syl3anc 1317 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
6059oveq1d 6541 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
6160oveq2d 6542 . . . . . . . . 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 2643 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6362adantr 479 . . . . . . 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 6533 . . . . . . . 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 480 . . . . . . 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 9850 . . . . . . . . . . . . 13 1 ∈ ℂ
67 pncan 10138 . . . . . . . . . . . . 13 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
6856, 66, 67sylancl 692 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
6968oveq2d 6542 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
7069oveq2d 6542 . . . . . . . . . 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 11177 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
7233, 71sylan 486 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
73 nn0p1nn 11181 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
7473adantl 480 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
7574nnzd 11315 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
76 bcpasc 12927 . . . . . . . . . . 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 690 . . . . . . . . . 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 2645 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
79 nn0p1nn 11181 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
80 nnnn0addcl 11172 . . . . . . . . . . . . . 14 (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8179, 80sylan 486 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8281nnnn0d 11200 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
83 bccl 12928 . . . . . . . . . . . 12 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8482, 75, 83syl2anc 690 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8584nn0cnd 11202 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
86 nn0z 11235 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
8786adantl 480 . . . . . . . . . . . . 13 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
88 bccl 12928 . . . . . . . . . . . . 13 ((((𝑁 + 1) + 𝑘) ∈ ℕ0𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8971, 87, 88syl2anc 690 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9033, 89sylan 486 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9190nn0cnd 11202 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
9285, 91addcomd 10089 . . . . . . . . 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 10059 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
9453, 93syl 17 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
9594adantr 479 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
9695, 56, 57addassd 9918 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
9796oveq1d 6541 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9878, 92, 973eqtr3d 2651 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9998adantr 479 . . . . . . 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 2650 . . . . . 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 448 . . . . 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 449 . . . 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 29 . . 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 11306 . 2 (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
105104impcom 444 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5789  (class class class)co 6526  cc 9790  0cc0 9792  1c1 9793   + caddc 9795  cmin 10117  cn 10869  0cn0 11141  cz 11212  cuz 11521  ...cfz 12154  Ccbc 12908  Σcsu 14212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-n0 11142  df-z 11213  df-uz 11522  df-rp 11667  df-fz 12155  df-fzo 12292  df-seq 12621  df-exp 12680  df-fac 12880  df-bc 12909  df-hash 12937  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-clim 14015  df-sum 14213
This theorem is referenced by:  arisum  14379
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