![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > decrmanc | Structured version Visualization version GIF version |
Description: Perform a multiply-add of two numerals ๐ and ๐ against a fixed multiplicand ๐ (no carry). (Contributed by AV, 16-Sep-2021.) |
Ref | Expression |
---|---|
decrmanc.a | โข ๐ด โ โ0 |
decrmanc.b | โข ๐ต โ โ0 |
decrmanc.n | โข ๐ โ โ0 |
decrmanc.m | โข ๐ = ;๐ด๐ต |
decrmanc.p | โข ๐ โ โ0 |
decrmanc.e | โข (๐ด ยท ๐) = ๐ธ |
decrmanc.f | โข ((๐ต ยท ๐) + ๐) = ๐น |
Ref | Expression |
---|---|
decrmanc | โข ((๐ ยท ๐) + ๐) = ;๐ธ๐น |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decrmanc.a | . 2 โข ๐ด โ โ0 | |
2 | decrmanc.b | . 2 โข ๐ต โ โ0 | |
3 | 0nn0 12492 | . 2 โข 0 โ โ0 | |
4 | decrmanc.n | . 2 โข ๐ โ โ0 | |
5 | decrmanc.m | . 2 โข ๐ = ;๐ด๐ต | |
6 | 4 | dec0h 12704 | . 2 โข ๐ = ;0๐ |
7 | decrmanc.p | . 2 โข ๐ โ โ0 | |
8 | 1, 7 | nn0mulcli 12515 | . . . . 5 โข (๐ด ยท ๐) โ โ0 |
9 | 8 | nn0cni 12489 | . . . 4 โข (๐ด ยท ๐) โ โ |
10 | 9 | addridi 11406 | . . 3 โข ((๐ด ยท ๐) + 0) = (๐ด ยท ๐) |
11 | decrmanc.e | . . 3 โข (๐ด ยท ๐) = ๐ธ | |
12 | 10, 11 | eqtri 2759 | . 2 โข ((๐ด ยท ๐) + 0) = ๐ธ |
13 | decrmanc.f | . 2 โข ((๐ต ยท ๐) + ๐) = ๐น | |
14 | 1, 2, 3, 4, 5, 6, 7, 12, 13 | decma 12733 | 1 โข ((๐ ยท ๐) + ๐) = ;๐ธ๐น |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 โ wcel 2105 (class class class)co 7412 0cc0 11114 + caddc 11117 ยท cmul 11119 โ0cn0 12477 ;cdc 12682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-ltxr 11258 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-dec 12683 |
This theorem is referenced by: decmul1 12746 37prm 17059 2503lem1 17075 4001lem1 17079 4001lem2 17080 4001lem3 17081 log2ub 26691 decpmulnc 41502 |
Copyright terms: Public domain | W3C validator |