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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 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 2763 | . . 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 12734 | . . . 4 ⊢ ;𝐸𝐷 = ((;10 · 𝐸) + 𝐷) | |
13 | 11, 12 | eqtri 2763 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 𝐸) + 𝐷) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul1c 12780 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
15 | dfdec10 12734 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
16 | 14, 15 | eqtr4i 2766 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℕ0cn0 12524 ;cdc 12731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 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 12732 |
This theorem is referenced by: 2exp8 17123 2exp11 17124 2exp16 17125 prmlem2 17154 631prm 17161 1259lem1 17165 1259lem2 17166 1259lem3 17167 1259lem4 17168 1259prm 17170 2503lem1 17171 2503lem2 17172 2503prm 17174 4001lem1 17175 4001lem2 17176 4001prm 17179 log2ublem3 27006 log2ub 27007 ex-fac 30480 dpmul 32880 12lcm5e60 41990 60lcm7e420 41992 3exp7 42035 3lexlogpow5ineq1 42036 3lexlogpow5ineq5 42042 aks4d1p1 42058 235t711 42318 ex-decpmul 42319 sum9cubes 42659 resqrtvalex 43635 imsqrtvalex 43636 wallispi2lem2 46028 fmtno5lem1 47478 fmtno5lem2 47479 fmtno5lem3 47480 257prm 47486 fmtno4nprmfac193 47499 fmtno5faclem1 47504 fmtno5faclem2 47505 m11nprm 47526 11t31e341 47657 2exp340mod341 47658 |
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