![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12643 | . . 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 12628 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
7 | 5, 6 | eqtri 2765 | . . 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 12628 | . . . 4 โข ;๐ธ๐ท = ((;10 ยท ๐ธ) + ๐ท) | |
13 | 11, 12 | eqtri 2765 | . . 3 โข (๐ ยท ๐ต) = ((;10 ยท ๐ธ) + ๐ท) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul2c 12675 | . 2 โข (๐ ยท ๐) = ((;10 ยท ๐ถ) + ๐ท) |
15 | dfdec10 12628 | . 2 โข ;๐ถ๐ท = ((;10 ยท ๐ถ) + ๐ท) | |
16 | 14, 15 | eqtr4i 2768 | 1 โข (๐ ยท ๐) = ;๐ถ๐ท |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 ยท cmul 11063 โ0cn0 12420 ;cdc 12625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-sub 11394 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-dec 12626 |
This theorem is referenced by: decmulnc 12692 2exp8 16968 2exp16 16970 prmlem2 16999 37prm 17000 1259lem2 17011 1259lem3 17012 1259lem4 17013 1259prm 17015 2503lem1 17016 2503lem2 17017 2503prm 17019 4001lem1 17020 4001lem2 17021 4001lem3 17022 4001prm 17024 log2ublem3 26314 log2ub 26315 birthday 26320 dpmul 31811 420gcd8e4 40492 420lcm8e840 40497 3exp7 40539 3lexlogpow5ineq1 40540 3lexlogpow5ineq5 40546 aks4d1p1 40562 decpmulnc 40830 235t711 40834 ex-decpmul 40835 resqrtvalex 41991 imsqrtvalex 41992 257prm 45827 fmtno4prmfac 45838 fmtno4prmfac193 45839 fmtno4nprmfac193 45840 m11nprm 45867 2exp340mod341 45999 |
Copyright terms: Public domain | W3C validator |