![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nnmulcld | GIF version |
Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
nnmulcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
Ref | Expression |
---|---|
nnmulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnmulcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
3 | nnmulcl 8953 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 (class class class)co 5888 · cmul 7829 ℕcn 8932 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-1rid 7931 ax-cnre 7935 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 df-inn 8933 |
This theorem is referenced by: qbtwnre 10270 bcval 10742 bcm1k 10753 bcp1n 10754 permnn 10764 cvg1nlemcxze 11004 cvg1nlemf 11005 cvg1nlemcau 11006 cvg1nlemres 11007 trireciplem 11521 efaddlem 11695 eftlub 11711 eirraplem 11797 modmulconst 11843 lcmval 12076 oddpwdclemxy 12182 oddpwdclemdc 12186 sqpweven 12188 2sqpwodd 12189 crth 12237 phimullem 12238 modprm0 12267 pcqmul 12316 pcaddlem 12351 pcbc 12362 oddprmdvds 12365 pockthlem 12367 pockthg 12368 evenennn 12407 lgseisenlem2 14722 2sqlem6 14738 |
Copyright terms: Public domain | W3C validator |