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Mirrors > Home > MPE Home > Th. List > numsucc | Structured version Visualization version GIF version |
Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numsucc.1 | โข ๐ โ โ0 |
numsucc.2 | โข ๐ = (๐ + 1) |
numsucc.3 | โข ๐ด โ โ0 |
numsucc.4 | โข (๐ด + 1) = ๐ต |
numsucc.5 | โข ๐ = ((๐ ยท ๐ด) + ๐) |
Ref | Expression |
---|---|
numsucc | โข (๐ + 1) = ((๐ ยท ๐ต) + 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numsucc.2 | . . . . . . 7 โข ๐ = (๐ + 1) | |
2 | numsucc.1 | . . . . . . . 8 โข ๐ โ โ0 | |
3 | 1nn0 12524 | . . . . . . . 8 โข 1 โ โ0 | |
4 | 2, 3 | nn0addcli 12545 | . . . . . . 7 โข (๐ + 1) โ โ0 |
5 | 1, 4 | eqeltri 2824 | . . . . . 6 โข ๐ โ โ0 |
6 | 5 | nn0cni 12520 | . . . . 5 โข ๐ โ โ |
7 | 6 | mulridi 11254 | . . . 4 โข (๐ ยท 1) = ๐ |
8 | 7 | oveq2i 7435 | . . 3 โข ((๐ ยท ๐ด) + (๐ ยท 1)) = ((๐ ยท ๐ด) + ๐) |
9 | numsucc.3 | . . . . 5 โข ๐ด โ โ0 | |
10 | 9 | nn0cni 12520 | . . . 4 โข ๐ด โ โ |
11 | ax-1cn 11202 | . . . 4 โข 1 โ โ | |
12 | 6, 10, 11 | adddii 11262 | . . 3 โข (๐ ยท (๐ด + 1)) = ((๐ ยท ๐ด) + (๐ ยท 1)) |
13 | 1 | eqcomi 2736 | . . . 4 โข (๐ + 1) = ๐ |
14 | numsucc.5 | . . . 4 โข ๐ = ((๐ ยท ๐ด) + ๐) | |
15 | 5, 9, 2, 13, 14 | numsuc 12727 | . . 3 โข (๐ + 1) = ((๐ ยท ๐ด) + ๐) |
16 | 8, 12, 15 | 3eqtr4ri 2766 | . 2 โข (๐ + 1) = (๐ ยท (๐ด + 1)) |
17 | numsucc.4 | . . 3 โข (๐ด + 1) = ๐ต | |
18 | 17 | oveq2i 7435 | . 2 โข (๐ ยท (๐ด + 1)) = (๐ ยท ๐ต) |
19 | 9, 3 | nn0addcli 12545 | . . . 4 โข (๐ด + 1) โ โ0 |
20 | 17, 19 | eqeltrri 2825 | . . 3 โข ๐ต โ โ0 |
21 | 5, 20 | num0u 12724 | . 2 โข (๐ ยท ๐ต) = ((๐ ยท ๐ต) + 0) |
22 | 16, 18, 21 | 3eqtri 2759 | 1 โข (๐ + 1) = ((๐ ยท ๐ต) + 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 (class class class)co 7424 0cc0 11144 1c1 11145 + caddc 11147 ยท cmul 11149 โ0cn0 12508 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-ltxr 11289 df-nn 12249 df-n0 12509 |
This theorem is referenced by: decsucc 12754 |
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