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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 12717 | . . 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 12702 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
7 | 5, 6 | eqtri 2755 | . . 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 12702 | . . . 4 โข ;๐ธ๐ท = ((;10 ยท ๐ธ) + ๐ท) | |
13 | 11, 12 | eqtri 2755 | . . 3 โข (๐ ยท ๐ต) = ((;10 ยท ๐ธ) + ๐ท) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul2c 12749 | . 2 โข (๐ ยท ๐) = ((;10 ยท ๐ถ) + ๐ท) |
15 | dfdec10 12702 | . 2 โข ;๐ถ๐ท = ((;10 ยท ๐ถ) + ๐ท) | |
16 | 14, 15 | eqtr4i 2758 | 1 โข (๐ ยท ๐) = ;๐ถ๐ท |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 โ wcel 2099 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 ยท cmul 11135 โ0cn0 12494 ;cdc 12699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-ltxr 11275 df-sub 11468 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-dec 12700 |
This theorem is referenced by: decmulnc 12766 2exp8 17049 2exp16 17051 prmlem2 17080 37prm 17081 1259lem2 17092 1259lem3 17093 1259lem4 17094 1259prm 17096 2503lem1 17097 2503lem2 17098 2503prm 17100 4001lem1 17101 4001lem2 17102 4001lem3 17103 4001prm 17105 log2ublem3 26867 log2ub 26868 birthday 26873 dpmul 32618 420gcd8e4 41414 420lcm8e840 41419 3exp7 41461 3lexlogpow5ineq1 41462 3lexlogpow5ineq5 41468 aks4d1p1 41484 decpmulnc 41783 235t711 41789 ex-decpmul 41790 resqrtvalex 42998 imsqrtvalex 42999 257prm 46824 fmtno4prmfac 46835 fmtno4prmfac193 46836 fmtno4nprmfac193 46837 m11nprm 46864 2exp340mod341 46996 |
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