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Mirrors > Home > ILE Home > Th. List > nnaddcld | 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 7715 | . 2 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A + B) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 391 | 1 ⊢ (φ → (A + B) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 (class class class)co 5455 + caddc 6714 ℕcn 7695 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-addrcl 6780 ax-addass 6785 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-br 3756 df-iota 4810 df-fv 4853 df-ov 5458 df-inn 7696 |
This theorem is referenced by: (None) |
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