![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nnmulcli | GIF version |
Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nnmulcli.1 | ⊢ 𝐴 ∈ ℕ |
nnmulcli.2 | ⊢ 𝐵 ∈ ℕ |
Ref | Expression |
---|---|
nnmulcli | ⊢ (𝐴 · 𝐵) ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnmulcli.1 | . 2 ⊢ 𝐴 ∈ ℕ | |
2 | nnmulcli.2 | . 2 ⊢ 𝐵 ∈ ℕ | |
3 | nnmulcl 8765 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ (𝐴 · 𝐵) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 (class class class)co 5782 · cmul 7649 ℕcn 8744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-1rid 7751 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-inn 8745 |
This theorem is referenced by: numnncl2 9228 ef01bndlem 11499 |
Copyright terms: Public domain | W3C validator |