Theorem nnaddcl 9162

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 6025	. . . . 5 ⊢ (𝑥 = 1 → (𝐴 + 𝑥) = (𝐴 + 1))
2	1	eleq1d 2300	. . . 4 ⊢ (𝑥 = 1 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ))
3	2	imbi2d 230	. . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)))
4		oveq2 6025	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
5	4	eleq1d 2300	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝑦) ∈ ℕ))
6	5	imbi2d 230	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ)))
7		oveq2 6025	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 1)))
8	7	eleq1d 2300	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
9	8	imbi2d 230	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
10		oveq2 6025	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
11	10	eleq1d 2300	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝐵) ∈ ℕ))
12	11	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ)))
13		peano2nn 9154	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
14		peano2nn 9154	. . . . . 6 ⊢ ((𝐴 + 𝑦) ∈ ℕ → ((𝐴 + 𝑦) + 1) ∈ ℕ)
15		nncn 9150	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
16		nncn 9150	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17		ax-1cn 8124	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		addass 8161	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
19	17, 18	mp3an3 1362	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
20	15, 16, 19	syl2an 289	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1)))
21	20	eleq1d 2300	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (((𝐴 + 𝑦) + 1) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ))
22	14, 21	imbitrid 154	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ))
23	22	expcom 116	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝐴 ∈ ℕ → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
24	23	a2d 26	. . 3 ⊢ (𝑦 ∈ ℕ → ((𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ) → (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)))
25	3, 6, 9, 12, 13, 24	nnind 9158	. 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ))
26	25	impcom 125	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  (class class class)co 6017  ℂcc 8029  1c1 8032   + caddc 8034  ℕcn 9142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-addrcl 8128  ax-addass 8133

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-inn 9143

This theorem is referenced by:  nnmulcl  9163  nn2ge  9175  nnaddcld  9190  nnnn0addcl  9431  nn0addcl  9436  9p1e10  9612  pythagtriplem4  12840  mulgnndir  13737