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Mirrors > Home > MPE Home > Th. List > decmul1 | Structured version Visualization version GIF version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) Remove hypothesis ๐ท โ โ0. (Revised by Steven Nguyen, 7-Dec-2022.) |
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.c | โข (๐ด ยท ๐) = ๐ถ |
decmul1.d | โข (๐ต ยท ๐) = ๐ท |
Ref | Expression |
---|---|
decmul1 | โข (๐ ยท ๐) = ;๐ถ๐ท |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decmul1.n | . . . 4 โข ๐ = ;๐ด๐ต | |
2 | decmul1.a | . . . . 5 โข ๐ด โ โ0 | |
3 | decmul1.b | . . . . 5 โข ๐ต โ โ0 | |
4 | 2, 3 | deccl 12565 | . . . 4 โข ;๐ด๐ต โ โ0 |
5 | 1, 4 | eqeltri 2834 | . . 3 โข ๐ โ โ0 |
6 | decmul1.p | . . 3 โข ๐ โ โ0 | |
7 | 5, 6 | num0u 12561 | . 2 โข (๐ ยท ๐) = ((๐ ยท ๐) + 0) |
8 | 0nn0 12361 | . . 3 โข 0 โ โ0 | |
9 | decmul1.c | . . 3 โข (๐ด ยท ๐) = ๐ถ | |
10 | 3, 6 | nn0mulcli 12384 | . . . . . 6 โข (๐ต ยท ๐) โ โ0 |
11 | 10 | nn0cni 12358 | . . . . 5 โข (๐ต ยท ๐) โ โ |
12 | 11 | addid1i 11275 | . . . 4 โข ((๐ต ยท ๐) + 0) = (๐ต ยท ๐) |
13 | decmul1.d | . . . 4 โข (๐ต ยท ๐) = ๐ท | |
14 | 12, 13 | eqtri 2765 | . . 3 โข ((๐ต ยท ๐) + 0) = ๐ท |
15 | 2, 3, 8, 1, 6, 9, 14 | decrmanc 12607 | . 2 โข ((๐ ยท ๐) + 0) = ;๐ถ๐ท |
16 | 7, 15 | eqtri 2765 | 1 โข (๐ ยท ๐) = ;๐ถ๐ท |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 โ wcel 2106 (class class class)co 7349 0cc0 10984 + caddc 10987 ยท cmul 10989 โ0cn0 12346 ;cdc 12550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-ov 7352 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-ltxr 11127 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-dec 12551 |
This theorem is referenced by: 2exp7 16894 37prm 16927 1259lem3 16939 1259lem4 16940 2503lem1 16943 2503lem2 16944 4001lem1 16947 4001lem2 16948 4001lem3 16949 4001prm 16951 log2ublem3 26220 log2ub 26221 bpos1 26553 ex-prmo 29201 dpmul 31563 60gcd6e6 40356 decpmulnc 40669 sqdeccom12 40671 ex-decpmul 40674 fmtno5lem3 45496 fmtno4prmfac193 45514 fmtno4nprmfac193 45515 fmtno5faclem1 45520 fmtno5faclem2 45521 |
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