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Mirrors > Home > MPE Home > Th. List > decmul1c | Structured version Visualization version GIF version |
Description: The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (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 |
decmul1c.e | ⊢ 𝐸 ∈ ℕ0 |
decmul1c.c | ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
decmul1c.2 | ⊢ (𝐵 · 𝑃) = ;𝐸𝐷 |
Ref | Expression |
---|---|
decmul1c | ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12747 | . . 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 12732 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2754 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | decmul1c.e | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
10 | decmul1c.c | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 | |
11 | decmul1c.2 | . . . 4 ⊢ (𝐵 · 𝑃) = ;𝐸𝐷 | |
12 | dfdec10 12732 | . . . 4 ⊢ ;𝐸𝐷 = ((;10 · 𝐸) + 𝐷) | |
13 | 11, 12 | eqtri 2754 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 𝐸) + 𝐷) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul1c 12778 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
15 | dfdec10 12732 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
16 | 14, 15 | eqtr4i 2757 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7424 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 ℕ0cn0 12524 ;cdc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12730 |
This theorem is referenced by: 2exp8 17091 2exp11 17092 2exp16 17093 prmlem2 17122 631prm 17129 1259lem1 17133 1259lem2 17134 1259lem3 17135 1259lem4 17136 1259prm 17138 2503lem1 17139 2503lem2 17140 2503prm 17142 4001lem1 17143 4001lem2 17144 4001prm 17147 log2ublem3 26976 log2ub 26977 ex-fac 30384 dpmul 32774 12lcm5e60 41707 60lcm7e420 41709 3exp7 41752 3lexlogpow5ineq1 41753 3lexlogpow5ineq5 41759 aks4d1p1 41775 235t711 42106 ex-decpmul 42107 sum9cubes 42326 resqrtvalex 43312 imsqrtvalex 43313 wallispi2lem2 45693 fmtno5lem1 47125 fmtno5lem2 47126 fmtno5lem3 47127 257prm 47133 fmtno4nprmfac193 47146 fmtno5faclem1 47151 fmtno5faclem2 47152 m11nprm 47173 11t31e341 47304 2exp340mod341 47305 |
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