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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 7936 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ |
4 | 3 | nncni 7924 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 7155 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5, 3 | eqeltri 2110 | 1 ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 (class class class)co 5512 0cc0 6889 + caddc 6892 · cmul 6894 ℕcn 7914 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-1rid 6991 ax-0id 6992 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-inn 7915 |
This theorem is referenced by: decnncl2 8385 |
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