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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 11659 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 (class class class)co 7155 + caddc 10539 ℕcn 11637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-1cn 10594 ax-addcl 10596 ax-addass 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-nn 11638 |
This theorem is referenced by: relexpaddnn 14409 pythagtriplem4 16155 pythagtriplem6 16157 pythagtriplem7 16158 pythagtriplem11 16161 pythagtriplem13 16163 pythagtriplem15 16165 vdwlem1 16316 vdwlem3 16318 vdwlem5 16320 vdwlem6 16321 vdwlem8 16323 vdwlem9 16324 vdwlem10 16325 vdwlem11 16326 prmgaplem2 16385 prmgaplcmlem2 16387 gsumsgrpccat 18003 gsumccatOLD 18004 aaliou3lem8 24933 lgsqrlem2 25922 lgseisenlem2 25951 2sqmod 26011 mdetlap 31097 ballotlem5 31757 faclimlem1 32975 faclimlem2 32976 faclim2 32980 nnadddir 39161 fmtnoprmfac2 43728 gbowpos 43923 |
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