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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 12569 | . . 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 12554 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
7 | 5, 6 | eqtri 2766 | . . 3 โข ๐ = ((;10 ยท ๐ด) + ๐ต) |
8 | 1, 2, 3, 4, 7 | numsuc 12565 | . 2 โข (๐ + 1) = ((;10 ยท ๐ด) + ๐ถ) |
9 | dfdec10 12554 | . 2 โข ;๐ด๐ถ = ((;10 ยท ๐ด) + ๐ถ) | |
10 | 8, 9 | eqtr4i 2769 | 1 โข (๐ + 1) = ;๐ด๐ถ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7350 0cc0 10985 1c1 10986 + caddc 10988 ยท cmul 10990 โ0cn0 12347 ;cdc 12551 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-ltxr 11128 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-dec 12552 |
This theorem is referenced by: 6p5lem 12621 dec2dvds 16870 13prm 16923 19prm 16925 37prm 16928 43prm 16929 139prm 16931 163prm 16932 317prm 16933 1259lem1 16938 1259lem3 16940 1259lem4 16941 1259lem5 16942 2503lem1 16944 2503lem2 16945 2503lem3 16946 2503prm 16947 4001lem1 16948 4001lem2 16949 4001lem3 16950 4001lem4 16951 4001prm 16952 log2ublem3 26220 log2ub 26221 birthday 26226 ex-exp 29180 dpmul4 31552 hgt750lem2 33026 420gcd8e4 40349 3lexlogpow5ineq1 40397 aks4d1p1 40419 fmtno2 45442 fmtno3 45443 fmtno4 45444 fmtno5lem1 45445 fmtno5lem2 45446 fmtno5lem3 45447 fmtno5lem4 45448 fmtno5 45449 257prm 45453 fmtno4prmfac 45464 fmtno4nprmfac193 45466 fmtno5fac 45474 139prmALT 45488 m7prm 45492 m11nprm 45493 2exp340mod341 45625 8exp8mod9 45628 ackval41 46481 |
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