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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.) |
Ref | Expression |
---|---|
decmul1.p | ⊢ 𝑃 ∈ ℕ0 |
decmul1.a | ⊢ 𝐴 ∈ ℕ0 |
decmul1.b | ⊢ 𝐵 ∈ ℕ0 |
decmul1.n | ⊢ 𝑁 = ;𝐴𝐵 |
decmul1.0 | ⊢ 𝐷 ∈ ℕ0 |
decmul1.c | ⊢ (𝐴 · 𝑃) = 𝐶 |
decmul1.d | ⊢ (𝐵 · 𝑃) = 𝐷 |
Ref | Expression |
---|---|
decmul1 | ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 11554 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decmul1.p | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
3 | decmul1.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | decmul1.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | decmul1.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
6 | dfdec10 11535 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2673 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | 0nn0 11345 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 3, 2 | nn0mulcli 11369 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 11342 | . . . . 5 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 10261 | . . . 4 ⊢ ((𝐴 · 𝑃) + 0) = (𝐴 · 𝑃) |
13 | decmul1.c | . . . 4 ⊢ (𝐴 · 𝑃) = 𝐶 | |
14 | 12, 13 | eqtri 2673 | . . 3 ⊢ ((𝐴 · 𝑃) + 0) = 𝐶 |
15 | decmul1.d | . . . . 5 ⊢ (𝐵 · 𝑃) = 𝐷 | |
16 | 15 | oveq2i 6701 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (0 + 𝐷) |
17 | 4, 2 | nn0mulcli 11369 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
18 | 17 | nn0cni 11342 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
19 | 18 | addid2i 10262 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (𝐵 · 𝑃) |
20 | 1 | nn0cni 11342 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
21 | 20 | mul01i 10264 | . . . . . 6 ⊢ (;10 · 0) = 0 |
22 | 21 | eqcomi 2660 | . . . . 5 ⊢ 0 = (;10 · 0) |
23 | 22 | oveq1i 6700 | . . . 4 ⊢ (0 + 𝐷) = ((;10 · 0) + 𝐷) |
24 | 16, 19, 23 | 3eqtr3i 2681 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 0) + 𝐷) |
25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 11600 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
26 | dfdec10 11535 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
27 | 25, 26 | eqtr4i 2676 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 ℕ0cn0 11330 ;cdc 11531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-dec 11532 |
This theorem is referenced by: sq10 13088 37prm 15875 1259lem3 15887 1259lem4 15888 2503lem1 15891 2503lem2 15892 4001lem1 15895 4001lem2 15896 4001lem3 15897 4001prm 15899 log2ublem3 24720 log2ub 24721 bpos1 25053 ex-prmo 27446 dpmul 29749 fmtno5lem3 41792 fmtno4prmfac193 41810 fmtno4nprmfac193 41811 fmtno5faclem1 41816 fmtno5faclem2 41817 2exp7 41839 |
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