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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpmul10 | Structured version Visualization version GIF version |
Description: Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dpval2.a | โข ๐ด โ โ0 |
dpval2.b | โข ๐ต โ โ |
Ref | Expression |
---|---|
dpmul10 | โข ((๐ด.๐ต) ยท ;10) = ;๐ด๐ต |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpval2.b | . . . . 5 โข ๐ต โ โ | |
2 | 1 | recni 11266 | . . . 4 โข ๐ต โ โ |
3 | 10nn 12731 | . . . . 5 โข ;10 โ โ | |
4 | 3 | nncni 12260 | . . . 4 โข ;10 โ โ |
5 | 3 | nnne0i 12290 | . . . 4 โข ;10 โ 0 |
6 | 2, 4, 5 | divcan2i 11995 | . . 3 โข (;10 ยท (๐ต / ;10)) = ๐ต |
7 | 6 | oveq2i 7437 | . 2 โข ((;10 ยท ๐ด) + (;10 ยท (๐ต / ;10))) = ((;10 ยท ๐ด) + ๐ต) |
8 | dpval2.a | . . . . 5 โข ๐ด โ โ0 | |
9 | 8, 1 | dpval2 32637 | . . . 4 โข (๐ด.๐ต) = (๐ด + (๐ต / ;10)) |
10 | 9 | oveq2i 7437 | . . 3 โข (;10 ยท (๐ด.๐ต)) = (;10 ยท (๐ด + (๐ต / ;10))) |
11 | dpcl 32635 | . . . . . 6 โข ((๐ด โ โ0 โง ๐ต โ โ) โ (๐ด.๐ต) โ โ) | |
12 | 8, 1, 11 | mp2an 690 | . . . . 5 โข (๐ด.๐ต) โ โ |
13 | 12 | recni 11266 | . . . 4 โข (๐ด.๐ต) โ โ |
14 | 4, 13 | mulcomi 11260 | . . 3 โข (;10 ยท (๐ด.๐ต)) = ((๐ด.๐ต) ยท ;10) |
15 | 8 | nn0cni 12522 | . . . 4 โข ๐ด โ โ |
16 | 2, 4, 5 | divcli 11994 | . . . 4 โข (๐ต / ;10) โ โ |
17 | 4, 15, 16 | adddii 11264 | . . 3 โข (;10 ยท (๐ด + (๐ต / ;10))) = ((;10 ยท ๐ด) + (;10 ยท (๐ต / ;10))) |
18 | 10, 14, 17 | 3eqtr3i 2764 | . 2 โข ((๐ด.๐ต) ยท ;10) = ((;10 ยท ๐ด) + (;10 ยท (๐ต / ;10))) |
19 | dfdec10 12718 | . 2 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
20 | 7, 18, 19 | 3eqtr4i 2766 | 1 โข ((๐ด.๐ต) ยท ;10) = ;๐ด๐ต |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 (class class class)co 7426 โcr 11145 0cc0 11146 1c1 11147 + caddc 11149 ยท cmul 11151 / cdiv 11909 โ0cn0 12510 ;cdc 12715 .cdp 32632 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-dec 12716 df-dp2 32616 df-dp 32633 |
This theorem is referenced by: decdiv10 32640 dpmul100 32641 dp3mul10 32642 dpmul1000 32643 dpmul 32657 dpmul4 32658 |
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