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Mirrors > Home > ILE Home > Th. List > numnncl2 | GIF version |
Description: Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
numnncl2.1 | ⊢ 𝑇 ∈ ℕ |
numnncl2.2 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
numnncl2 | ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numnncl2.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ | |
2 | numnncl2.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 8742 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ |
4 | 3 | nncni 8730 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 7904 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5, 3 | eqeltri 2212 | 1 ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 (class class class)co 5774 0cc0 7620 + caddc 7623 · cmul 7625 ℕcn 8720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-0id 7728 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 |
This theorem is referenced by: decnncl2 9205 |
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