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Mirrors > Home > ILE Home > Th. List > decnncl2 | GIF version |
Description: Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decnncl2.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
decnncl2 | ⊢ ;𝐴0 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 8480 | . 2 ⊢ ;𝐴0 = ((;10 · 𝐴) + 0) | |
2 | 10nn 8492 | . . 3 ⊢ ;10 ∈ ℕ | |
3 | decnncl2.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
4 | 2, 3 | numnncl2 8499 | . 2 ⊢ ((;10 · 𝐴) + 0) ∈ ℕ |
5 | 1, 4 | eqeltri 2151 | 1 ⊢ ;𝐴0 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 ℕcn 8039 ;cdc 8477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-1rid 7083 ax-0id 7084 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-dec 8478 |
This theorem is referenced by: 3dec 9642 |
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