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Mirrors > Home > MPE Home > Th. List > nn0mulcli | Structured version Visualization version GIF version |
Description: Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nn0addcli.1 | ⊢ 𝑀 ∈ ℕ0 |
nn0addcli.2 | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0mulcli | ⊢ (𝑀 · 𝑁) ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0addcli.1 | . 2 ⊢ 𝑀 ∈ ℕ0 | |
2 | nn0addcli.2 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
3 | nn0mulcl 12504 | . 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝑀 · 𝑁) ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 (class class class)co 7404 · cmul 11111 ℕ0cn0 12468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-nn 12209 df-n0 12469 |
This theorem is referenced by: numnncl 12683 num0u 12684 numcl 12686 numsuc 12687 numlt 12698 decle 12707 decrmanc 12730 decsubi 12736 decmul1 12737 decmulnc 12740 decmul10add 12742 expnass 14168 nn0opthlem1 14224 faclbnd4lem1 14249 dec2dvds 16992 dec5dvds 16993 gcdi 17002 decexp2 17004 decsplit 17012 log2ublem1 26431 log2ublem2 26432 log2ublem3 26433 log2ub 26434 bclbnd 26763 dpmul 32057 sqn5i 41147 decpmulnc 41149 decpmul 41150 sqdeccom12 41151 |
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