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Mirrors > Home > MPE Home > Th. List > nnaddcld | Structured version Visualization version GIF version |
Description: Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
nnmulcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
Ref | Expression |
---|---|
nnaddcld | ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnmulcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
3 | nnaddcl 12286 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 (class class class)co 7430 + caddc 11155 ℕcn 12263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 ax-1cn 11210 ax-addcl 11212 ax-addass 11217 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 |
This theorem is referenced by: relexpaddnn 15086 pythagtriplem4 16852 pythagtriplem6 16854 pythagtriplem7 16855 pythagtriplem11 16858 pythagtriplem13 16860 pythagtriplem15 16862 vdwlem1 17014 vdwlem3 17016 vdwlem5 17018 vdwlem6 17019 vdwlem8 17021 vdwlem9 17022 vdwlem10 17023 vdwlem11 17024 prmgaplem2 17083 prmgaplcmlem2 17085 gsumsgrpccat 18865 aaliou3lem8 26401 lgsqrlem2 27405 lgseisenlem2 27434 2sqmod 27494 mdetlap 33792 ballotlem5 34480 faclimlem1 35722 faclimlem2 35723 faclim2 35727 lcmineqlem22 42031 nnadddir 42283 zaddcom 42458 fimgmcyc 42520 flt4lem6 42644 fmtnoprmfac2 47491 gbowpos 47683 |
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