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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 8600 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ |
4 | 3 | nncni 8588 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 7775 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5, 3 | eqeltri 2172 | 1 ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 (class class class)co 5706 0cc0 7500 + caddc 7503 · cmul 7505 ℕcn 8578 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-1rid 7602 ax-0id 7603 ax-cnre 7606 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-iota 5024 df-fv 5067 df-ov 5709 df-inn 8579 |
This theorem is referenced by: decnncl2 9057 |
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