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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 11648 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 (class class class)co 7135 + caddc 10529 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 ax-addcl 10586 ax-addass 10591 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 |
This theorem is referenced by: relexpaddnn 14402 pythagtriplem4 16146 pythagtriplem6 16148 pythagtriplem7 16149 pythagtriplem11 16152 pythagtriplem13 16154 pythagtriplem15 16156 vdwlem1 16307 vdwlem3 16309 vdwlem5 16311 vdwlem6 16312 vdwlem8 16314 vdwlem9 16315 vdwlem10 16316 vdwlem11 16317 prmgaplem2 16376 prmgaplcmlem2 16378 gsumsgrpccat 17996 gsumccatOLD 17997 aaliou3lem8 24941 lgsqrlem2 25931 lgseisenlem2 25960 2sqmod 26020 mdetlap 31185 ballotlem5 31867 faclimlem1 33088 faclimlem2 33089 faclim2 33093 lcmineqlem22 39338 nnadddir 39471 fmtnoprmfac2 44084 gbowpos 44277 |
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