![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > decrmac | Structured version Visualization version GIF version |
Description: Perform a multiply-add of two numerals ๐ and ๐ against a fixed multiplicand ๐ (with 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 |
decrmac.f | โข ๐น โ โ0 |
decrmac.g | โข ๐บ โ โ0 |
decrmac.e | โข ((๐ด ยท ๐) + ๐บ) = ๐ธ |
decrmac.2 | โข ((๐ต ยท ๐) + ๐) = ;๐บ๐น |
Ref | Expression |
---|---|
decrmac | 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 12525 | . 2 โข 0 โ โ0 | |
4 | decrmanc.n | . 2 โข ๐ โ โ0 | |
5 | decrmanc.m | . 2 โข ๐ = ;๐ด๐ต | |
6 | 4 | dec0h 12737 | . 2 โข ๐ = ;0๐ |
7 | decrmanc.p | . 2 โข ๐ โ โ0 | |
8 | decrmac.f | . 2 โข ๐น โ โ0 | |
9 | decrmac.g | . 2 โข ๐บ โ โ0 | |
10 | 9 | nn0cni 12522 | . . . . 5 โข ๐บ โ โ |
11 | 10 | addlidi 11440 | . . . 4 โข (0 + ๐บ) = ๐บ |
12 | 11 | oveq2i 7437 | . . 3 โข ((๐ด ยท ๐) + (0 + ๐บ)) = ((๐ด ยท ๐) + ๐บ) |
13 | decrmac.e | . . 3 โข ((๐ด ยท ๐) + ๐บ) = ๐ธ | |
14 | 12, 13 | eqtri 2756 | . 2 โข ((๐ด ยท ๐) + (0 + ๐บ)) = ๐ธ |
15 | decrmac.2 | . 2 โข ((๐ต ยท ๐) + ๐) = ;๐บ๐น | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | decmac 12767 | 1 โข ((๐ ยท ๐) + ๐) = ;๐ธ๐น |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 (class class class)co 7426 0cc0 11146 + caddc 11149 ยท cmul 11151 โ0cn0 12510 ;cdc 12715 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-sub 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-dec 12716 |
This theorem is referenced by: 2exp16 17067 139prm 17100 163prm 17101 1259lem1 17107 1259lem3 17109 1259lem4 17110 2503lem1 17113 2503lem2 17114 4001lem1 17117 4001lem3 17119 4001prm 17121 log2ub 26901 139prmALT 46965 127prm 46968 |
Copyright terms: Public domain | W3C validator |