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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 12566 | . . . 4 โข ;๐ด๐ต โ โ0 |
5 | 1, 4 | eqeltri 2835 | . . 3 โข ๐ โ โ0 |
6 | decmul1.p | . . 3 โข ๐ โ โ0 | |
7 | 5, 6 | num0u 12562 | . 2 โข (๐ ยท ๐) = ((๐ ยท ๐) + 0) |
8 | 0nn0 12362 | . . 3 โข 0 โ โ0 | |
9 | decmul1.c | . . 3 โข (๐ด ยท ๐) = ๐ถ | |
10 | 3, 6 | nn0mulcli 12385 | . . . . . 6 โข (๐ต ยท ๐) โ โ0 |
11 | 10 | nn0cni 12359 | . . . . 5 โข (๐ต ยท ๐) โ โ |
12 | 11 | addid1i 11276 | . . . 4 โข ((๐ต ยท ๐) + 0) = (๐ต ยท ๐) |
13 | decmul1.d | . . . 4 โข (๐ต ยท ๐) = ๐ท | |
14 | 12, 13 | eqtri 2766 | . . 3 โข ((๐ต ยท ๐) + 0) = ๐ท |
15 | 2, 3, 8, 1, 6, 9, 14 | decrmanc 12608 | . 2 โข ((๐ ยท ๐) + 0) = ;๐ถ๐ท |
16 | 7, 15 | eqtri 2766 | 1 โข (๐ ยท ๐) = ;๐ถ๐ท |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7350 0cc0 10985 + caddc 10988 ยท cmul 10990 โ0cn0 12347 ;cdc 12551 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-ltxr 11128 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-dec 12552 |
This theorem is referenced by: 2exp7 16895 37prm 16928 1259lem3 16940 1259lem4 16941 2503lem1 16944 2503lem2 16945 4001lem1 16948 4001lem2 16949 4001lem3 16950 4001prm 16952 log2ublem3 26220 log2ub 26221 bpos1 26553 ex-prmo 29189 dpmul 31551 60gcd6e6 40347 decpmulnc 40648 sqdeccom12 40650 ex-decpmul 40653 fmtno5lem3 45465 fmtno4prmfac193 45483 fmtno4nprmfac193 45484 fmtno5faclem1 45489 fmtno5faclem2 45490 |
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