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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 9083 | . 2 ⊢ ;𝐴0 = ((;10 · 𝐴) + 0) | |
2 | 10nn 9095 | . . 3 ⊢ ;10 ∈ ℕ | |
3 | decnncl2.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
4 | 2, 3 | numnncl2 9102 | . 2 ⊢ ((;10 · 𝐴) + 0) ∈ ℕ |
5 | 1, 4 | eqeltri 2185 | 1 ⊢ ;𝐴0 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1461 (class class class)co 5726 0cc0 7541 1c1 7542 + caddc 7544 · cmul 7546 ℕcn 8624 ;cdc 9080 |
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 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-1rid 7646 ax-0id 7647 ax-cnre 7650 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-iota 5044 df-fv 5087 df-ov 5729 df-inn 8625 df-2 8683 df-3 8684 df-4 8685 df-5 8686 df-6 8687 df-7 8688 df-8 8689 df-9 8690 df-dec 9081 |
This theorem is referenced by: 3dec 10348 |
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