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Mirrors > Home > MPE Home > Th. List > decmul1 | Structured version Visualization version Unicode version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul1.p | |
decmul1.a | |
decmul1.b | |
decmul1.n | ; |
decmul1.0 | |
decmul1.c | |
decmul1.d |
Ref | Expression |
---|---|
decmul1 | ; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 11516 | . . 3 ; | |
2 | decmul1.p | . . 3 | |
3 | decmul1.a | . . 3 | |
4 | decmul1.b | . . 3 | |
5 | decmul1.n | . . . 4 ; | |
6 | dfdec10 11497 | . . . 4 ; ; | |
7 | 5, 6 | eqtri 2644 | . . 3 ; |
8 | decmul1.0 | . . 3 | |
9 | 0nn0 11307 | . . 3 | |
10 | 3, 2 | nn0mulcli 11331 | . . . . . 6 |
11 | 10 | nn0cni 11304 | . . . . 5 |
12 | 11 | addid1i 10223 | . . . 4 |
13 | decmul1.c | . . . 4 | |
14 | 12, 13 | eqtri 2644 | . . 3 |
15 | decmul1.d | . . . . 5 | |
16 | 15 | oveq2i 6661 | . . . 4 |
17 | 4, 2 | nn0mulcli 11331 | . . . . . 6 |
18 | 17 | nn0cni 11304 | . . . . 5 |
19 | 18 | addid2i 10224 | . . . 4 |
20 | 1 | nn0cni 11304 | . . . . . . 7 ; |
21 | 20 | mul01i 10226 | . . . . . 6 ; |
22 | 21 | eqcomi 2631 | . . . . 5 ; |
23 | 22 | oveq1i 6660 | . . . 4 ; |
24 | 16, 19, 23 | 3eqtr3i 2652 | . . 3 ; |
25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 11562 | . 2 ; |
26 | dfdec10 11497 | . 2 ; ; | |
27 | 25, 26 | eqtr4i 2647 | 1 ; |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 (class class class)co 6650 cc0 9936 c1 9937 caddc 9939 cmul 9941 cn0 11292 ;cdc 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 |
This theorem is referenced by: sq10 13048 37prm 15828 1259lem3 15840 1259lem4 15841 2503lem1 15844 2503lem2 15845 4001lem1 15848 4001lem2 15849 4001lem3 15850 4001prm 15852 log2ublem3 24675 log2ub 24676 bpos1 25008 ex-prmo 27316 dpmul 29621 fmtno5lem3 41467 fmtno4prmfac193 41485 fmtno4nprmfac193 41486 fmtno5faclem1 41491 fmtno5faclem2 41492 2exp7 41514 |
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