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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 8971 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴 · 𝐵) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 (class class class)co 5897 · cmul 7847 ℕcn 8950 |
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 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-1rid 7949 ax-cnre 7953 |
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 5900 df-inn 8951 |
This theorem is referenced by: numnncl2 9437 ef01bndlem 11799 pockthi 12393 lgsdir2lem5 14911 |
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