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Mirrors > Home > MPE Home > Th. List > numaddc | Structured version Visualization version GIF version |
Description: Add two decimal integers 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numma.1 | ⊢ 𝑇 ∈ ℕ0 |
numma.2 | ⊢ 𝐴 ∈ ℕ0 |
numma.3 | ⊢ 𝐵 ∈ ℕ0 |
numma.4 | ⊢ 𝐶 ∈ ℕ0 |
numma.5 | ⊢ 𝐷 ∈ ℕ0 |
numma.6 | ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) |
numma.7 | ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) |
numaddc.8 | ⊢ 𝐹 ∈ ℕ0 |
numaddc.9 | ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 |
numaddc.10 | ⊢ (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹) |
Ref | Expression |
---|---|
numaddc | ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.6 | . . . . . 6 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
2 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
3 | numma.2 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
4 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
5 | 2, 3, 4 | numcl 11702 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2835 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
7 | 6 | nn0cni 11496 | . . . 4 ⊢ 𝑀 ∈ ℂ |
8 | 7 | mulid1i 10234 | . . 3 ⊢ (𝑀 · 1) = 𝑀 |
9 | 8 | oveq1i 6823 | . 2 ⊢ ((𝑀 · 1) + 𝑁) = (𝑀 + 𝑁) |
10 | numma.4 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
11 | numma.5 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
12 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
13 | 1nn0 11500 | . . 3 ⊢ 1 ∈ ℕ0 | |
14 | numaddc.8 | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
15 | 3 | nn0cni 11496 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
16 | 15 | mulid1i 10234 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
17 | 16 | oveq1i 6823 | . . . 4 ⊢ ((𝐴 · 1) + (𝐶 + 1)) = (𝐴 + (𝐶 + 1)) |
18 | 10 | nn0cni 11496 | . . . . 5 ⊢ 𝐶 ∈ ℂ |
19 | ax-1cn 10186 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | 15, 18, 19 | addassi 10240 | . . . 4 ⊢ ((𝐴 + 𝐶) + 1) = (𝐴 + (𝐶 + 1)) |
21 | numaddc.9 | . . . 4 ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 | |
22 | 17, 20, 21 | 3eqtr2i 2788 | . . 3 ⊢ ((𝐴 · 1) + (𝐶 + 1)) = 𝐸 |
23 | 4 | nn0cni 11496 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
24 | 23 | mulid1i 10234 | . . . . 5 ⊢ (𝐵 · 1) = 𝐵 |
25 | 24 | oveq1i 6823 | . . . 4 ⊢ ((𝐵 · 1) + 𝐷) = (𝐵 + 𝐷) |
26 | numaddc.10 | . . . 4 ⊢ (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹) | |
27 | 25, 26 | eqtri 2782 | . . 3 ⊢ ((𝐵 · 1) + 𝐷) = ((𝑇 · 1) + 𝐹) |
28 | 2, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27 | nummac 11750 | . 2 ⊢ ((𝑀 · 1) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
29 | 9, 28 | eqtr3i 2784 | 1 ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 (class class class)co 6813 1c1 10129 + caddc 10131 · cmul 10133 ℕ0cn0 11484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-ltxr 10271 df-sub 10460 df-nn 11213 df-n0 11485 |
This theorem is referenced by: decaddc 11764 decaddcOLD 11765 |
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