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Mirrors > Home > MPE Home > Th. List > decnncl2 | Structured version Visualization version GIF version |
Description: Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decnncl2.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
decnncl2 | ⊢ ;𝐴0 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 12439 | . 2 ⊢ ;𝐴0 = ((;10 · 𝐴) + 0) | |
2 | 10nn 12452 | . . 3 ⊢ ;10 ∈ ℕ | |
3 | decnncl2.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
4 | 2, 3 | numnncl2 12459 | . 2 ⊢ ((;10 · 𝐴) + 0) ∈ ℕ |
5 | 1, 4 | eqeltri 2837 | 1 ⊢ ;𝐴0 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 (class class class)co 7271 0cc0 10872 1c1 10873 + caddc 10875 · cmul 10877 ℕcn 11973 ;cdc 12436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-ltxr 11015 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-dec 12437 |
This theorem is referenced by: 43prm 16821 4001prm 16844 dpadd3 31182 dpmul 31183 dpmul4 31184 hgt750lem 32627 60gcd6e6 40009 60gcd7e1 40010 420gcd8e4 40011 12lcm5e60 40013 60lcm6e60 40014 60lcm7e420 40015 420lcm8e840 40016 lcmineqlem 40057 fmtno4nprmfac193 44995 tgoldbach 45238 |
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