decmul1c - Metamath Proof Explorer

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Theorem decmul1c 12767

Description: The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)

**Assertion**
Ref	Expression
decmul1c	⊢ (𝑁 · 𝑃) = ;𝐶𝐷

**Proof of Theorem decmul1c**
Step	Hyp	Ref	Expression
1		10nn0 12720	. . 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 12705	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
7	5, 6	eqtri 2756	. . 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 12705	. . . 4 ⊢ ;𝐸𝐷 = ((;10 · 𝐸) + 𝐷)
13	11, 12	eqtri 2756	. . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 𝐸) + 𝐷)
14	1, 2, 3, 4, 7, 8, 9, 10, 13	nummul1c 12751	. 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷)
15		dfdec10 12705	. 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷)
16	14, 15	eqtr4i 2759	1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7415 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 ℕ₀cn0 12497 cdc 12702

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 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-ltxr 11278 df-sub 11471 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-dec 12703

This theorem is referenced by: 2exp8 17052 2exp11 17053 2exp16 17054 prmlem2 17083 631prm 17090 1259lem1 17094 1259lem2 17095 1259lem3 17096 1259lem4 17097 1259prm 17099 2503lem1 17100 2503lem2 17101 2503prm 17103 4001lem1 17104 4001lem2 17105 4001prm 17108 log2ublem3 26874 log2ub 26875 ex-fac 30255 dpmul 32631 12lcm5e60 41474 60lcm7e420 41476 3exp7 41519 3lexlogpow5ineq1 41520 3lexlogpow5ineq5 41526 aks4d1p1 41542 235t711 41858 ex-decpmul 41859 sum9cubes 42087 resqrtvalex 43066 imsqrtvalex 43067 wallispi2lem2 45451 fmtno5lem1 46884 fmtno5lem2 46885 fmtno5lem3 46886 257prm 46892 fmtno4nprmfac193 46905 fmtno5faclem1 46910 fmtno5faclem2 46911 m11nprm 46932 11t31e341 47063 2exp340mod341 47064

	
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