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Mirrors > Home > MPE Home > Th. List > decsuc | Structured version Visualization version GIF version |
Description: The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
declt.a | โข ๐ด โ โ0 |
declt.b | โข ๐ต โ โ0 |
decsuc.c | โข (๐ต + 1) = ๐ถ |
decsuc.n | โข ๐ = ;๐ด๐ต |
Ref | Expression |
---|---|
decsuc | โข (๐ + 1) = ;๐ด๐ถ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12568 | . . 3 โข ;10 โ โ0 | |
2 | declt.a | . . 3 โข ๐ด โ โ0 | |
3 | declt.b | . . 3 โข ๐ต โ โ0 | |
4 | decsuc.c | . . 3 โข (๐ต + 1) = ๐ถ | |
5 | decsuc.n | . . . 4 โข ๐ = ;๐ด๐ต | |
6 | dfdec10 12553 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
7 | 5, 6 | eqtri 2765 | . . 3 โข ๐ = ((;10 ยท ๐ด) + ๐ต) |
8 | 1, 2, 3, 4, 7 | numsuc 12564 | . 2 โข (๐ + 1) = ((;10 ยท ๐ด) + ๐ถ) |
9 | dfdec10 12553 | . 2 โข ;๐ด๐ถ = ((;10 ยท ๐ด) + ๐ถ) | |
10 | 8, 9 | eqtr4i 2768 | 1 โข (๐ + 1) = ;๐ด๐ถ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 โ wcel 2106 (class class class)co 7349 0cc0 10984 1c1 10985 + caddc 10987 ยท cmul 10989 โ0cn0 12346 ;cdc 12550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-ov 7352 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-ltxr 11127 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-dec 12551 |
This theorem is referenced by: 6p5lem 12620 dec2dvds 16869 13prm 16922 19prm 16924 37prm 16927 43prm 16928 139prm 16930 163prm 16931 317prm 16932 1259lem1 16937 1259lem3 16939 1259lem4 16940 1259lem5 16941 2503lem1 16943 2503lem2 16944 2503lem3 16945 2503prm 16946 4001lem1 16947 4001lem2 16948 4001lem3 16949 4001lem4 16950 4001prm 16951 log2ublem3 26220 log2ub 26221 birthday 26226 ex-exp 29192 dpmul4 31564 hgt750lem2 33038 420gcd8e4 40358 3lexlogpow5ineq1 40406 aks4d1p1 40428 fmtno2 45491 fmtno3 45492 fmtno4 45493 fmtno5lem1 45494 fmtno5lem2 45495 fmtno5lem3 45496 fmtno5lem4 45497 fmtno5 45498 257prm 45502 fmtno4prmfac 45513 fmtno4nprmfac193 45515 fmtno5fac 45523 139prmALT 45537 m7prm 45541 m11nprm 45542 2exp340mod341 45674 8exp8mod9 45677 ackval41 46530 |
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