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Mirrors > Home > MPE Home > Th. List > decadd | Structured version Visualization version GIF version |
Description: Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decma.a | ⊢ 𝐴 ∈ ℕ0 |
decma.b | ⊢ 𝐵 ∈ ℕ0 |
decma.c | ⊢ 𝐶 ∈ ℕ0 |
decma.d | ⊢ 𝐷 ∈ ℕ0 |
decma.m | ⊢ 𝑀 = ;𝐴𝐵 |
decma.n | ⊢ 𝑁 = ;𝐶𝐷 |
decadd.e | ⊢ (𝐴 + 𝐶) = 𝐸 |
decadd.f | ⊢ (𝐵 + 𝐷) = 𝐹 |
Ref | Expression |
---|---|
decadd | ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12594 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decma.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | decma.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | decma.c | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | decma.d | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
6 | decma.m | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
7 | dfdec10 12579 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | 6, 7 | eqtri 2765 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
9 | decma.n | . . . 4 ⊢ 𝑁 = ;𝐶𝐷 | |
10 | dfdec10 12579 | . . . 4 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
11 | 9, 10 | eqtri 2765 | . . 3 ⊢ 𝑁 = ((;10 · 𝐶) + 𝐷) |
12 | decadd.e | . . 3 ⊢ (𝐴 + 𝐶) = 𝐸 | |
13 | decadd.f | . . 3 ⊢ (𝐵 + 𝐷) = 𝐹 | |
14 | 1, 2, 3, 4, 5, 8, 11, 12, 13 | numadd 12623 | . 2 ⊢ (𝑀 + 𝑁) = ((;10 · 𝐸) + 𝐹) |
15 | dfdec10 12579 | . 2 ⊢ ;𝐸𝐹 = ((;10 · 𝐸) + 𝐹) | |
16 | 14, 15 | eqtr4i 2768 | 1 ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7351 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ℕ0cn0 12371 ;cdc 12576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-dec 12577 |
This theorem is referenced by: decaddm10 12635 decaddi 12636 10p10e20 12671 dec5dvds2 16897 2exp16 16923 37prm 16953 43prm 16954 317prm 16958 631prm 16959 1259lem2 16964 1259lem3 16965 1259lem4 16966 2503lem1 16969 2503lem2 16970 4001lem1 16973 4001lem2 16974 4001lem3 16975 log2ublem3 26250 log2ub 26251 1kp2ke3k 29219 hgt750lemd 33073 hgt750lem2 33077 12gcd5e1 40398 3lexlogpow5ineq1 40449 decpmul 40711 sqdeccom12 40712 sq3deccom12 40713 ex-decpmul 40715 resqrtvalex 41828 imsqrtvalex 41829 fmtno5lem4 45649 257prm 45654 fmtno4prmfac 45665 fmtno4nprmfac193 45667 fmtno5faclem3 45674 fmtno5fac 45675 |
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