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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 11934 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) ∈ ℕ0) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7156 · cmul 10542 ℕ0cn0 11898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-nn 11639 df-n0 11899 |
This theorem is referenced by: quoremnn0ALT 13226 expmulz 13476 faclbnd4lem3 13656 oddge22np1 15698 mulgcd 15896 rpmulgcd2 16000 hashgcdlem 16125 odzdvds 16132 prmreclem3 16254 vdwapf 16308 vdwlem5 16321 vdwlem6 16322 smndex2dbas 18079 odmodnn0 18668 odmulg 18683 odadd 18970 ablfacrplem 19187 ablfacrp2 19189 2lgslem1c 25969 2lgslem3a 25972 2lgslem3b 25973 2lgslem3c 25974 2lgslem3d 25975 dchrisumlem1 26065 eulerpartlemsv2 31616 eulerpartlemsf 31617 eulerpartlems 31618 eulerpartlemv 31622 eulerpartlemb 31626 breprexplemc 31903 erdsze2lem1 32450 erdsze2lem2 32451 3cubeslem3l 39332 3cubeslem3r 39333 pell1qrge1 39516 jm2.27c 39653 rmxdiophlem 39661 stoweidlem1 42335 wallispilem4 42402 wallispilem5 42403 wallispi2lem2 42406 stirlinglem3 42410 stirlinglem5 42412 stirlinglem7 42414 stirlinglem10 42417 stirlinglem11 42418 etransclem32 42600 etransclem44 42612 etransclem46 42614 fmtnofac2lem 43779 fmtnofac1 43781 2pwp1prm 43800 lighneallem3 43821 fppr2odd 43945 ply1mulgsumlem2 44490 |
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