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Theorem bcxmas 14687
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 14686 . . . . 5 (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2 oveq2 6773 . . . . . 6 (𝑚 = 0 → (0...𝑚) = (0...0))
32sumeq1d 14551 . . . . 5 (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
41, 3eqeq12d 2739 . . . 4 (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
54imbi2d 329 . . 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 14686 . . . . 5 (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
7 oveq2 6773 . . . . . 6 (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
87sumeq1d 14551 . . . . 5 (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
96, 8eqeq12d 2739 . . . 4 (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
109imbi2d 329 . . 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 14686 . . . . 5 (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
12 oveq2 6773 . . . . . 6 (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
1312sumeq1d 14551 . . . . 5 (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
1411, 13eqeq12d 2739 . . . 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 329 . . 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 14686 . . . . 5 (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
17 oveq2 6773 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
1817sumeq1d 14551 . . . . 5 (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
1916, 18eqeq12d 2739 . . . 4 (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
2019imbi2d 329 . . 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 11420 . . . . 5 0 ∈ ℕ0
22 nn0addcl 11441 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
23 bcn0 13212 . . . . . 6 ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
2422, 23syl 17 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
2521, 24mpan2 709 . . . 4 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
26 0z 11501 . . . . 5 0 ∈ ℤ
27 1nn0 11421 . . . . . . 7 1 ∈ ℕ0
2825, 27syl6eqel 2811 . . . . . 6 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
2928nn0cnd 11466 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
30 bcxmaslem1 14686 . . . . . 6 (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3130fsum1 14596 . . . . 5 ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
3226, 29, 31sylancr 698 . . . 4 (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
33 peano2nn0 11446 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
34 nn0addcl 11441 . . . . . 6 (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
3533, 21, 34sylancl 697 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
36 bcn0 13212 . . . . 5 (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3735, 36syl 17 . . . 4 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3825, 32, 373eqtr4rd 2769 . . 3 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
39 simpr 479 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40 elnn0uz 11839 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ (ℤ‘0))
4139, 40sylib 208 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ‘0))
42 simpl 474 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
43 elfznn0 12547 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
44 nn0addcl 11441 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
4542, 43, 44syl2an 495 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
46 elfzelz 12456 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
4746adantl 473 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
48 bccl 13224 . . . . . . . . . . . 12 (((𝑁 + 𝑗) ∈ ℕ0𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
4945, 47, 48syl2anc 696 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
5049nn0cnd 11466 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
51 bcxmaslem1 14686 . . . . . . . . . 10 (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
5241, 50, 51fsump1 14607 . . . . . . . . 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 11415 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
5453adantr 472 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
55 nn0cn 11415 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
5655adantl 473 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
57 1cnd 10169 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 1 ∈ ℂ)
58 add32r 10368 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
5954, 56, 57, 58syl3anc 1439 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
6059oveq1d 6780 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
6160oveq2d 6781 . . . . . . . . 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 2758 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6362adantr 472 . . . . . . 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 6772 . . . . . . . 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 473 . . . . . . 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 10107 . . . . . . . . . . . . 13 1 ∈ ℂ
67 pncan 10400 . . . . . . . . . . . . 13 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
6856, 66, 67sylancl 697 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
6968oveq2d 6781 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
7069oveq2d 6781 . . . . . . . . . 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 11441 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
7233, 71sylan 489 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
73 nn0p1nn 11445 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
7473adantl 473 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
7574nnzd 11594 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
76 bcpasc 13223 . . . . . . . . . . 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 696 . . . . . . . . . 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 2760 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
79 nn0p1nn 11445 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
80 nnnn0addcl 11436 . . . . . . . . . . . . . 14 (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8179, 80sylan 489 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
8281nnnn0d 11464 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
83 bccl 13224 . . . . . . . . . . . 12 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8482, 75, 83syl2anc 696 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8584nn0cnd 11466 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
86 nn0z 11513 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
8786adantl 473 . . . . . . . . . . . . 13 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
88 bccl 13224 . . . . . . . . . . . . 13 ((((𝑁 + 1) + 𝑘) ∈ ℕ0𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8971, 87, 88syl2anc 696 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9033, 89sylan 489 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
9190nn0cnd 11466 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
9285, 91addcomd 10351 . . . . . . . . 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 10321 . . . . . . . . . . . . 13 (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
9453, 93syl 17 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
9594adantr 472 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
9695, 56, 57addassd 10175 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
9796oveq1d 6780 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9878, 92, 973eqtr3d 2766 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9998adantr 472 . . . . . . 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 2765 . . . . . 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 449 . . . . 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 450 . . . 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 11585 . 2 (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
105104impcom 445 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 383   = wceq 1596  wcel 2103  cfv 6001  (class class class)co 6765  cc 10047  0cc0 10049  1c1 10050   + caddc 10052  cmin 10379  cn 11133  0cn0 11405  cz 11490  cuz 11800  ...cfz 12440  Ccbc 13204  Σcsu 14536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-fac 13176  df-bc 13205  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-sum 14537
This theorem is referenced by:  arisum  14712
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