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Mirrors > Home > MPE Home > Th. List > decmul2c | Structured version Visualization version GIF version |
Description: The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul1.p | โข ๐ โ โ0 |
decmul1.a | โข ๐ด โ โ0 |
decmul1.b | โข ๐ต โ โ0 |
decmul1.n | โข ๐ = ;๐ด๐ต |
decmul1.0 | โข ๐ท โ โ0 |
decmul1c.e | โข ๐ธ โ โ0 |
decmul2c.c | โข ((๐ ยท ๐ด) + ๐ธ) = ๐ถ |
decmul2c.2 | โข (๐ ยท ๐ต) = ;๐ธ๐ท |
Ref | Expression |
---|---|
decmul2c | โข (๐ ยท ๐) = ;๐ถ๐ท |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12697 | . . 3 โข ;10 โ โ0 | |
2 | decmul1.p | . . 3 โข ๐ โ โ0 | |
3 | decmul1.a | . . 3 โข ๐ด โ โ0 | |
4 | decmul1.b | . . 3 โข ๐ต โ โ0 | |
5 | decmul1.n | . . . 4 โข ๐ = ;๐ด๐ต | |
6 | dfdec10 12682 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
7 | 5, 6 | eqtri 2760 | . . 3 โข ๐ = ((;10 ยท ๐ด) + ๐ต) |
8 | decmul1.0 | . . 3 โข ๐ท โ โ0 | |
9 | decmul1c.e | . . 3 โข ๐ธ โ โ0 | |
10 | decmul2c.c | . . 3 โข ((๐ ยท ๐ด) + ๐ธ) = ๐ถ | |
11 | decmul2c.2 | . . . 4 โข (๐ ยท ๐ต) = ;๐ธ๐ท | |
12 | dfdec10 12682 | . . . 4 โข ;๐ธ๐ท = ((;10 ยท ๐ธ) + ๐ท) | |
13 | 11, 12 | eqtri 2760 | . . 3 โข (๐ ยท ๐ต) = ((;10 ยท ๐ธ) + ๐ท) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul2c 12729 | . 2 โข (๐ ยท ๐) = ((;10 ยท ๐ถ) + ๐ท) |
15 | dfdec10 12682 | . 2 โข ;๐ถ๐ท = ((;10 ยท ๐ถ) + ๐ท) | |
16 | 14, 15 | eqtr4i 2763 | 1 โข (๐ ยท ๐) = ;๐ถ๐ท |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 โ wcel 2106 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 ยท cmul 11117 โ0cn0 12474 ;cdc 12679 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-ltxr 11255 df-sub 11448 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-dec 12680 |
This theorem is referenced by: decmulnc 12746 2exp8 17024 2exp16 17026 prmlem2 17055 37prm 17056 1259lem2 17067 1259lem3 17068 1259lem4 17069 1259prm 17071 2503lem1 17072 2503lem2 17073 2503prm 17075 4001lem1 17076 4001lem2 17077 4001lem3 17078 4001prm 17080 log2ublem3 26460 log2ub 26461 birthday 26466 dpmul 32117 420gcd8e4 40957 420lcm8e840 40962 3exp7 41004 3lexlogpow5ineq1 41005 3lexlogpow5ineq5 41011 aks4d1p1 41027 decpmulnc 41281 235t711 41287 ex-decpmul 41288 resqrtvalex 42478 imsqrtvalex 42479 257prm 46308 fmtno4prmfac 46319 fmtno4prmfac193 46320 fmtno4nprmfac193 46321 m11nprm 46348 2exp340mod341 46480 |
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