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Mirrors > Home > MPE Home > Th. List > decmul2c | 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 |
decmul2c.c | ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 |
decmul2c.2 | ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 |
Ref | Expression |
---|---|
decmul2c | ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12749 | . . 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 12734 | . . . 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 | decmul2c.c | . . 3 ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 | |
11 | decmul2c.2 | . . . 4 ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 | |
12 | dfdec10 12734 | . . . 4 ⊢ ;𝐸𝐷 = ((;10 · 𝐸) + 𝐷) | |
13 | 11, 12 | eqtri 2754 | . . 3 ⊢ (𝑃 · 𝐵) = ((;10 · 𝐸) + 𝐷) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul2c 12781 | . 2 ⊢ (𝑃 · 𝑁) = ((;10 · 𝐶) + 𝐷) |
15 | dfdec10 12734 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
16 | 14, 15 | eqtr4i 2757 | 1 ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7426 0cc0 11160 1c1 11161 + caddc 11163 · cmul 11165 ℕ0cn0 12526 ;cdc 12731 |
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 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 |
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 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-ltxr 11305 df-sub 11498 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12732 |
This theorem is referenced by: decmulnc 12798 2exp8 17093 2exp16 17095 prmlem2 17124 37prm 17125 1259lem2 17136 1259lem3 17137 1259lem4 17138 1259prm 17140 2503lem1 17141 2503lem2 17142 2503prm 17144 4001lem1 17145 4001lem2 17146 4001lem3 17147 4001prm 17149 log2ublem3 26979 log2ub 26980 birthday 26985 dpmul 32776 420gcd8e4 41707 420lcm8e840 41712 3exp7 41754 3lexlogpow5ineq1 41755 3lexlogpow5ineq5 41761 aks4d1p1 41777 decpmulnc 42006 235t711 42012 ex-decpmul 42013 resqrtvalex 43330 imsqrtvalex 43331 257prm 47151 fmtno4prmfac 47162 fmtno4prmfac193 47163 fmtno4nprmfac193 47164 m11nprm 47191 2exp340mod341 47323 |
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