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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 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) |
| Colors of variables: wff setvar class |
| 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 |
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