Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nn0mulcld | Structured version Visualization version GIF version |
Description: Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
nn0addcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0mulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | nn0addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
3 | nn0mulcl 11206 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) ∈ ℕ0) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 (class class class)co 6549 · cmul 9820 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-n0 11170 |
This theorem is referenced by: quoremnn0ALT 12518 expmulz 12768 faclbnd4lem3 12944 oddge22np1 14911 mulgcd 15103 rpmulgcd2 15208 hashgcdlem 15331 odzdvds 15338 prmreclem3 15460 vdwapf 15514 vdwlem5 15527 vdwlem6 15528 odmodnn0 17782 odmulg 17796 odadd 18076 ablfacrplem 18287 ablfacrp2 18289 2lgslem1c 24918 2lgslem3a 24921 2lgslem3b 24922 2lgslem3c 24923 2lgslem3d 24924 dchrisumlem1 24978 eulerpartlemsv2 29747 eulerpartlemsf 29748 eulerpartlems 29749 eulerpartlemv 29753 eulerpartlemb 29757 erdsze2lem1 30439 erdsze2lem2 30440 pell1qrge1 36452 jm2.27c 36592 rmxdiophlem 36600 stoweidlem1 38894 wallispilem4 38961 wallispilem5 38962 wallispi2lem2 38965 stirlinglem3 38969 stirlinglem5 38971 stirlinglem7 38973 stirlinglem10 38976 stirlinglem11 38977 etransclem32 39159 etransclem44 39171 etransclem46 39173 fmtnofac2lem 40018 fmtnofac1 40020 2pwp1prm 40041 lighneallem3 40062 ply1mulgsumlem2 41969 |
Copyright terms: Public domain | W3C validator |