Theorem nnaddcl 12289

Description: Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Assertion

Ref	Expression
nnaddcl	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)

Proof of Theorem nnaddcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7439	. . . . 5 ⊢ (𝑥 = 1 → (𝐴 + 𝑥) = (𝐴 + 1))
2	1	eleq1d 2826	. . . 4 ⊢ (𝑥 = 1 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)))
4		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
5	4	eleq1d 2826	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝑦) ∈ ℕ))
6	5	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ)))
7		oveq2 7439	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 1)))
8	7	eleq1d 2826	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
10		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
11	10	eleq1d 2826	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝐵) ∈ ℕ))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ)))
13		peano2nn 12278	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
14		peano2nn 12278	. . . . . 6 ⊢ ((𝐴 + 𝑦) ∈ ℕ → ((𝐴 + 𝑦) + 1) ∈ ℕ)
15		nncn 12274	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
16		nncn 12274	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17		ax-1cn 11213	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		addass 11242	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
19	17, 18	mp3an3 1452	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
20	15, 16, 19	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
21	20	eleq1d 2826	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (((𝐴 + 𝑦) + 1) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
22	14, 21	imbitrid 244	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ))
23	22	expcom 413	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝐴 ∈ ℕ → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
24	23	a2d 29	. . 3 ⊢ (𝑦 ∈ ℕ → ((𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ) → (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
25	3, 6, 9, 12, 13, 24	nnind 12284	. 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ))
26	25	impcom 407	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  (class class class)co 7431  ℂcc 11153  1c1 11156   + caddc 11158  ℕcn 12266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-1cn 11213  ax-addcl 11215  ax-addass 11220

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267

This theorem is referenced by:  nnmulcl  12290  nnaddcld  12318  nnnn0addcl  12556  nn0addcl  12561  zaddcl  12657  9p1e10  12735  pythagtriplem4  16857  vdwapun  17012  vdwap1  17015  vdwlem2  17020  prmgaplem7  17095  prmgapprmolem  17099  mulgnndir  19121  uniioombllem3  25620  ballotlem1  34489  ballotlem2  34491  ballotlemfmpn  34497  ballotlem4  34501  ballotlemimin  34508  ballotlemsdom  34514  ballotlemsel1i  34515  ballotlemfrceq  34531  ballotlemfrcn0  34532  ballotlem1ri  34537  ballotth  34540  nndivsub  36458  nnadddir  42305  gbepos  47745  gbowpos  47746  nnsgrpmgm  48092