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Theorem nnaddcl 7675
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 B ℕ) → (A + B) ℕ)

Proof of Theorem nnaddcl
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5463 . . . . 5 (x = 1 → (A + x) = (A + 1))
21eleq1d 2103 . . . 4 (x = 1 → ((A + x) ℕ ↔ (A + 1) ℕ))
32imbi2d 219 . . 3 (x = 1 → ((A ℕ → (A + x) ℕ) ↔ (A ℕ → (A + 1) ℕ)))
4 oveq2 5463 . . . . 5 (x = y → (A + x) = (A + y))
54eleq1d 2103 . . . 4 (x = y → ((A + x) ℕ ↔ (A + y) ℕ))
65imbi2d 219 . . 3 (x = y → ((A ℕ → (A + x) ℕ) ↔ (A ℕ → (A + y) ℕ)))
7 oveq2 5463 . . . . 5 (x = (y + 1) → (A + x) = (A + (y + 1)))
87eleq1d 2103 . . . 4 (x = (y + 1) → ((A + x) ℕ ↔ (A + (y + 1)) ℕ))
98imbi2d 219 . . 3 (x = (y + 1) → ((A ℕ → (A + x) ℕ) ↔ (A ℕ → (A + (y + 1)) ℕ)))
10 oveq2 5463 . . . . 5 (x = B → (A + x) = (A + B))
1110eleq1d 2103 . . . 4 (x = B → ((A + x) ℕ ↔ (A + B) ℕ))
1211imbi2d 219 . . 3 (x = B → ((A ℕ → (A + x) ℕ) ↔ (A ℕ → (A + B) ℕ)))
13 peano2nn 7667 . . 3 (A ℕ → (A + 1) ℕ)
14 peano2nn 7667 . . . . . 6 ((A + y) ℕ → ((A + y) + 1) ℕ)
15 nncn 7663 . . . . . . . 8 (A ℕ → A ℂ)
16 nncn 7663 . . . . . . . 8 (y ℕ → y ℂ)
17 ax-1cn 6736 . . . . . . . . 9 1
18 addass 6769 . . . . . . . . 9 ((A y 1 ℂ) → ((A + y) + 1) = (A + (y + 1)))
1917, 18mp3an3 1220 . . . . . . . 8 ((A y ℂ) → ((A + y) + 1) = (A + (y + 1)))
2015, 16, 19syl2an 273 . . . . . . 7 ((A y ℕ) → ((A + y) + 1) = (A + (y + 1)))
2120eleq1d 2103 . . . . . 6 ((A y ℕ) → (((A + y) + 1) ℕ ↔ (A + (y + 1)) ℕ))
2214, 21syl5ib 143 . . . . 5 ((A y ℕ) → ((A + y) ℕ → (A + (y + 1)) ℕ))
2322expcom 109 . . . 4 (y ℕ → (A ℕ → ((A + y) ℕ → (A + (y + 1)) ℕ)))
2423a2d 23 . . 3 (y ℕ → ((A ℕ → (A + y) ℕ) → (A ℕ → (A + (y + 1)) ℕ)))
253, 6, 9, 12, 13, 24nnind 7671 . 2 (B ℕ → (A ℕ → (A + B) ℕ))
2625impcom 116 1 ((A B ℕ) → (A + B) ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  (class class class)co 5455  cc 6669  1c1 6672   + caddc 6674  cn 7655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6734  ax-resscn 6735  ax-1cn 6736  ax-1re 6737  ax-addrcl 6740  ax-addass 6745
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7656
This theorem is referenced by:  nnmulcl  7676  nn2ge  7687  nnaddcld  7701  nnnn0addcl  7948  nn0addcl  7953
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