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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 𝐷 ∈ ℕ0. (Revised by Steven Nguyen, 7-Dec-2022.) |
Ref | Expression |
---|---|
decmul1.p | ⊢ 𝑃 ∈ ℕ0 |
decmul1.a | ⊢ 𝐴 ∈ ℕ0 |
decmul1.b | ⊢ 𝐵 ∈ ℕ0 |
decmul1.n | ⊢ 𝑁 = ;𝐴𝐵 |
decmul1.c | ⊢ (𝐴 · 𝑃) = 𝐶 |
decmul1.d | ⊢ (𝐵 · 𝑃) = 𝐷 |
Ref | Expression |
---|---|
decmul1 | ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decmul1.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
2 | decmul1.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | decmul1.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
4 | 2, 3 | deccl 12187 | . . . 4 ⊢ ;𝐴𝐵 ∈ ℕ0 |
5 | 1, 4 | eqeltri 2829 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
6 | decmul1.p | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
7 | 5, 6 | num0u 12183 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
8 | 0nn0 11984 | . . 3 ⊢ 0 ∈ ℕ0 | |
9 | decmul1.c | . . 3 ⊢ (𝐴 · 𝑃) = 𝐶 | |
10 | 3, 6 | nn0mulcli 12007 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 11981 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 10898 | . . . 4 ⊢ ((𝐵 · 𝑃) + 0) = (𝐵 · 𝑃) |
13 | decmul1.d | . . . 4 ⊢ (𝐵 · 𝑃) = 𝐷 | |
14 | 12, 13 | eqtri 2761 | . . 3 ⊢ ((𝐵 · 𝑃) + 0) = 𝐷 |
15 | 2, 3, 8, 1, 6, 9, 14 | decrmanc 12229 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ;𝐶𝐷 |
16 | 7, 15 | eqtri 2761 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2113 (class class class)co 7164 0cc0 10608 + caddc 10611 · cmul 10613 ℕ0cn0 11969 ;cdc 12172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-ltxr 10751 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-dec 12173 |
This theorem is referenced by: 2exp7 16517 37prm 16550 1259lem3 16562 1259lem4 16563 2503lem1 16566 2503lem2 16567 4001lem1 16570 4001lem2 16571 4001lem3 16572 4001prm 16574 log2ublem3 25678 log2ub 25679 bpos1 26011 ex-prmo 28388 dpmul 30754 60gcd6e6 39621 decpmulnc 39875 sqdeccom12 39877 ex-decpmul 39880 fmtno5lem3 44525 fmtno4prmfac193 44543 fmtno4nprmfac193 44544 fmtno5faclem1 44549 fmtno5faclem2 44550 |
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