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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0dp2dp | Structured version Visualization version GIF version |
Description: Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
0dp2dp.a | โข ๐ด โ โ0 |
0dp2dp.b | โข ๐ต โ โ+ |
Ref | Expression |
---|---|
0dp2dp | โข ((0._๐ด๐ต) ยท ;10) = (๐ด.๐ต) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0dp2dp.a | . . . 4 โข ๐ด โ โ0 | |
2 | 0dp2dp.b | . . . 4 โข ๐ต โ โ+ | |
3 | 0p1e1 12370 | . . . 4 โข (0 + 1) = 1 | |
4 | 0z 12605 | . . . 4 โข 0 โ โค | |
5 | 1z 12628 | . . . 4 โข 1 โ โค | |
6 | 1, 2, 3, 4, 5 | dpexpp1 32649 | . . 3 โข ((๐ด.๐ต) ยท (;10โ0)) = ((0._๐ด๐ต) ยท (;10โ1)) |
7 | 10nn0 12731 | . . . . . 6 โข ;10 โ โ0 | |
8 | 7 | nn0cni 12520 | . . . . 5 โข ;10 โ โ |
9 | exp0 14068 | . . . . 5 โข (;10 โ โ โ (;10โ0) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . 4 โข (;10โ0) = 1 |
11 | 10 | oveq2i 7435 | . . 3 โข ((๐ด.๐ต) ยท (;10โ0)) = ((๐ด.๐ต) ยท 1) |
12 | exp1 14070 | . . . . 5 โข (;10 โ โ โ (;10โ1) = ;10) | |
13 | 8, 12 | ax-mp 5 | . . . 4 โข (;10โ1) = ;10 |
14 | 13 | oveq2i 7435 | . . 3 โข ((0._๐ด๐ต) ยท (;10โ1)) = ((0._๐ด๐ต) ยท ;10) |
15 | 6, 11, 14 | 3eqtr3ri 2764 | . 2 โข ((0._๐ด๐ต) ยท ;10) = ((๐ด.๐ต) ยท 1) |
16 | 1, 2 | rpdpcl 32644 | . . . 4 โข (๐ด.๐ต) โ โ+ |
17 | rpcn 13022 | . . . 4 โข ((๐ด.๐ต) โ โ+ โ (๐ด.๐ต) โ โ) | |
18 | 16, 17 | ax-mp 5 | . . 3 โข (๐ด.๐ต) โ โ |
19 | mulrid 11248 | . . 3 โข ((๐ด.๐ต) โ โ โ ((๐ด.๐ต) ยท 1) = (๐ด.๐ต)) | |
20 | 18, 19 | ax-mp 5 | . 2 โข ((๐ด.๐ต) ยท 1) = (๐ด.๐ต) |
21 | 15, 20 | eqtri 2755 | 1 โข ((0._๐ด๐ต) ยท ;10) = (๐ด.๐ต) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 (class class class)co 7424 โcc 11142 0cc0 11144 1c1 11145 ยท cmul 11149 โ0cn0 12508 ;cdc 12713 โ+crp 13012 โcexp 14064 _cdp2 32612 .cdp 32629 |
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-cnex 11200 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 ax-pre-mulgt0 11221 |
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-rmo 3372 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-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 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-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-rp 13013 df-seq 14005 df-exp 14065 df-dp2 32613 df-dp 32630 |
This theorem is referenced by: hgt750lem 34288 |
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