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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 | ⊢ 𝐴 ∈ ℕ0 |
declt.b | ⊢ 𝐵 ∈ ℕ0 |
decsuc.c | ⊢ (𝐵 + 1) = 𝐶 |
decsuc.n | ⊢ 𝑁 = ;𝐴𝐵 |
Ref | Expression |
---|---|
decsuc | ⊢ (𝑁 + 1) = ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12635 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | declt.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | declt.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | decsuc.c | . . 3 ⊢ (𝐵 + 1) = 𝐶 | |
5 | decsuc.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
6 | dfdec10 12620 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2764 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | 1, 2, 3, 4, 7 | numsuc 12631 | . 2 ⊢ (𝑁 + 1) = ((;10 · 𝐴) + 𝐶) |
9 | dfdec10 12620 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
10 | 8, 9 | eqtr4i 2767 | 1 ⊢ (𝑁 + 1) = ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7356 0cc0 11050 1c1 11051 + caddc 11053 · cmul 11055 ℕ0cn0 12412 ;cdc 12617 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-ltxr 11193 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-dec 12618 |
This theorem is referenced by: 6p5lem 12687 dec2dvds 16934 13prm 16987 19prm 16989 37prm 16992 43prm 16993 139prm 16995 163prm 16996 317prm 16997 1259lem1 17002 1259lem3 17004 1259lem4 17005 1259lem5 17006 2503lem1 17008 2503lem2 17009 2503lem3 17010 2503prm 17011 4001lem1 17012 4001lem2 17013 4001lem3 17014 4001lem4 17015 4001prm 17016 log2ublem3 26296 log2ub 26297 birthday 26302 ex-exp 29341 dpmul4 31714 hgt750lem2 33205 420gcd8e4 40453 3lexlogpow5ineq1 40501 aks4d1p1 40523 fmtno2 45713 fmtno3 45714 fmtno4 45715 fmtno5lem1 45716 fmtno5lem2 45717 fmtno5lem3 45718 fmtno5lem4 45719 fmtno5 45720 257prm 45724 fmtno4prmfac 45735 fmtno4nprmfac193 45737 fmtno5fac 45745 139prmALT 45759 m7prm 45763 m11nprm 45764 2exp340mod341 45896 8exp8mod9 45899 ackval41 46752 |
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