Theorem nnaddcl 12188

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 7364	. . . . 5 ⊢ (𝑥 = 1 → (𝐴 + 𝑥) = (𝐴 + 1))
2	1	eleq1d 2824	. . . 4 ⊢ (𝑥 = 1 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ))
3	2	imbi2d 341	. . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)))
4		oveq2 7364	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
5	4	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝑦) ∈ ℕ))
6	5	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ)))
7		oveq2 7364	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 1)))
8	7	eleq1d 2824	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
9	8	imbi2d 341	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
10		oveq2 7364	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
11	10	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝐵) ∈ ℕ))
12	11	imbi2d 341	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ)))
13		peano2nn 12177	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
14		peano2nn 12177	. . . . . 6 ⊢ ((𝐴 + 𝑦) ∈ ℕ → ((𝐴 + 𝑦) + 1) ∈ ℕ)
15		nncn 12173	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
16		nncn 12173	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17		ax-1cn 11087	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		addass 11116	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
19	17, 18	mp3an3 1458	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
20	15, 16, 19	syl2an 602	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
21	20	eleq1d 2824	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (((𝐴 + 𝑦) + 1) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
22	14, 21	imbitrid 245	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ))
23	22	expcom 414	. . . 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 12183	. 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ))
26	25	impcom 408	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  (class class class)co 7356  ℂcc 11027  1c1 11030   + caddc 11032  ℕcn 12165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-1cn 11087  ax-addcl 11089  ax-addass 11094

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12166

This theorem is referenced by:  nnmulcl  12189  nnaddcld  12220  nnadddir  12224  nnnn0addcl  12458  nn0addcl  12463  zaddcl  12558  9p1e10  12637  pythagtriplem4  16781  vdwapun  16936  vdwap1  16939  vdwlem2  16944  prmgaplem7  17019  prmgapprmolem  17023  mulgnndir  19070  uniioombllem3  25570  ballotlem1  34671  ballotlem2  34673  ballotlemfmpn  34679  ballotlem4  34683  ballotlemimin  34690  ballotlemsdom  34696  ballotlemsel1i  34697  ballotlemfrceq  34713  ballotlemfrcn0  34714  ballotlem1ri  34719  ballotth  34722  nndivsub  36685  gbepos  48249  gbowpos  48250  nnsgrpmgm  48667