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Mirrors > Home > MPE Home > Th. List > numsuc | Structured version Visualization version GIF version |
Description: The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numnncl.1 | โข ๐ โ โ0 |
numnncl.2 | โข ๐ด โ โ0 |
numcl.2 | โข ๐ต โ โ0 |
numsuc.4 | โข (๐ต + 1) = ๐ถ |
numsuc.5 | โข ๐ = ((๐ ยท ๐ด) + ๐ต) |
Ref | Expression |
---|---|
numsuc | โข (๐ + 1) = ((๐ ยท ๐ด) + ๐ถ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numsuc.5 | . . 3 โข ๐ = ((๐ ยท ๐ด) + ๐ต) | |
2 | 1 | oveq1i 7414 | . 2 โข (๐ + 1) = (((๐ ยท ๐ด) + ๐ต) + 1) |
3 | numnncl.1 | . . . . 5 โข ๐ โ โ0 | |
4 | numnncl.2 | . . . . 5 โข ๐ด โ โ0 | |
5 | 3, 4 | nn0mulcli 12511 | . . . 4 โข (๐ ยท ๐ด) โ โ0 |
6 | 5 | nn0cni 12485 | . . 3 โข (๐ ยท ๐ด) โ โ |
7 | numcl.2 | . . . 4 โข ๐ต โ โ0 | |
8 | 7 | nn0cni 12485 | . . 3 โข ๐ต โ โ |
9 | ax-1cn 11167 | . . 3 โข 1 โ โ | |
10 | 6, 8, 9 | addassi 11225 | . 2 โข (((๐ ยท ๐ด) + ๐ต) + 1) = ((๐ ยท ๐ด) + (๐ต + 1)) |
11 | numsuc.4 | . . 3 โข (๐ต + 1) = ๐ถ | |
12 | 11 | oveq2i 7415 | . 2 โข ((๐ ยท ๐ด) + (๐ต + 1)) = ((๐ ยท ๐ด) + ๐ถ) |
13 | 2, 10, 12 | 3eqtri 2758 | 1 โข (๐ + 1) = ((๐ ยท ๐ด) + ๐ถ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 (class class class)co 7404 1c1 11110 + caddc 11112 ยท cmul 11114 โ0cn0 12473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-nn 12214 df-n0 12474 |
This theorem is referenced by: decsuc 12709 numsucc 12718 decbin3 12820 |
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