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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 8970 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ |
4 | 3 | nncni 8958 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 8128 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5, 3 | eqeltri 2262 | 1 ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 (class class class)co 5895 0cc0 7840 + caddc 7843 · cmul 7845 ℕcn 8948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-1rid 7947 ax-0id 7948 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 df-inn 8949 |
This theorem is referenced by: decnncl2 9436 |
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