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Mirrors > Home > MPE Home > Th. List > nnaddcl | Structured version Visualization version GIF version |
Description: Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Ref | Expression |
---|---|
nnaddcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 1 → (𝐴 + 𝑥) = (𝐴 + 1)) | |
2 | 1 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 1 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 1) ∈ ℕ)) |
3 | 2 | imbi2d 329 | . . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ))) |
4 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦)) | |
5 | 4 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝑦) ∈ ℕ)) |
6 | 5 | imbi2d 329 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝑦) ∈ ℕ))) |
7 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 1))) | |
8 | 7 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ)) |
9 | 8 | imbi2d 329 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ))) |
10 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵)) | |
11 | 10 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) ∈ ℕ ↔ (𝐴 + 𝐵) ∈ ℕ)) |
12 | 11 | imbi2d 329 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 + 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ))) |
13 | peano2nn 10909 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
14 | peano2nn 10909 | . . . . . 6 ⊢ ((𝐴 + 𝑦) ∈ ℕ → ((𝐴 + 𝑦) + 1) ∈ ℕ) | |
15 | nncn 10905 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
16 | nncn 10905 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
17 | ax-1cn 9873 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
18 | addass 9902 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1))) | |
19 | 17, 18 | mp3an3 1405 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1))) |
20 | 15, 16, 19 | syl2an 493 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) + 1) = (𝐴 + (𝑦 + 1))) |
21 | 20 | eleq1d 2672 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (((𝐴 + 𝑦) + 1) ∈ ℕ ↔ (𝐴 + (𝑦 + 1)) ∈ ℕ)) |
22 | 14, 21 | syl5ib 233 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 + 𝑦) ∈ ℕ → (𝐴 + (𝑦 + 1)) ∈ ℕ)) |
23 | 22 | expcom 450 | . . . 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 10915 | . 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 + 𝐵) ∈ ℕ)) |
26 | 25 | impcom 445 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-addass 9880 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 |
This theorem is referenced by: nnmulcl 10920 nnaddcld 10944 nnnn0addcl 11200 nn0addcl 11205 zaddcl 11294 9p1e10 11372 pythagtriplem4 15362 vdwapun 15516 vdwap1 15519 vdwlem2 15524 prmgaplem7 15599 prmgapprmolem 15603 mulgnndir 17392 mulgnndirOLD 17393 uniioombllem3 23159 ballotlem1 29875 ballotlem2 29877 ballotlemfmpn 29883 ballotlem4 29887 ballotlemimin 29894 ballotlemsdom 29900 ballotlemsel1i 29901 ballotlemfrceq 29917 ballotlemfrcn0 29918 ballotlem1ri 29923 ballotth 29926 nndivsub 31626 gbepos 40180 gbopos 40181 av-numclwwlk2lem1 41532 nnsgrpmgm 41606 |
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