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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 12487 | . 2 โข 0 โ โ0 | |
4 | decrmanc.n | . 2 โข ๐ โ โ0 | |
5 | decrmanc.m | . 2 โข ๐ = ;๐ด๐ต | |
6 | 4 | dec0h 12699 | . 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 12484 | . . . . 5 โข ๐บ โ โ |
11 | 10 | addlidi 11402 | . . . 4 โข (0 + ๐บ) = ๐บ |
12 | 11 | oveq2i 7420 | . . 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 2761 | . 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 12729 | 1 โข ((๐ ยท ๐) + ๐) = ;๐ธ๐น |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7409 0cc0 11110 + caddc 11113 ยท cmul 11115 โ0cn0 12472 ;cdc 12677 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-dec 12678 |
This theorem is referenced by: 2exp16 17024 139prm 17057 163prm 17058 1259lem1 17064 1259lem3 17066 1259lem4 17067 2503lem1 17070 2503lem2 17071 4001lem1 17074 4001lem3 17076 4001prm 17078 log2ub 26454 139prmALT 46264 127prm 46267 |
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