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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 11949 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2877 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
7 | 6 | nn0cni 11746 | . . . 4 ⊢ 𝑀 ∈ ℂ |
8 | 7 | mulid1i 10480 | . . 3 ⊢ (𝑀 · 1) = 𝑀 |
9 | 8 | oveq1i 7017 | . 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 11750 | . . 3 ⊢ 1 ∈ ℕ0 | |
14 | numaddc.8 | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
15 | 3 | nn0cni 11746 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
16 | 15 | mulid1i 10480 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
17 | 16 | oveq1i 7017 | . . . 4 ⊢ ((𝐴 · 1) + (𝐶 + 1)) = (𝐴 + (𝐶 + 1)) |
18 | 10 | nn0cni 11746 | . . . . 5 ⊢ 𝐶 ∈ ℂ |
19 | ax-1cn 10430 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | 15, 18, 19 | addassi 10486 | . . . 4 ⊢ ((𝐴 + 𝐶) + 1) = (𝐴 + (𝐶 + 1)) |
21 | numaddc.9 | . . . 4 ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 | |
22 | 17, 20, 21 | 3eqtr2i 2823 | . . 3 ⊢ ((𝐴 · 1) + (𝐶 + 1)) = 𝐸 |
23 | 4 | nn0cni 11746 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
24 | 23 | mulid1i 10480 | . . . . 5 ⊢ (𝐵 · 1) = 𝐵 |
25 | 24 | oveq1i 7017 | . . . 4 ⊢ ((𝐵 · 1) + 𝐷) = (𝐵 + 𝐷) |
26 | numaddc.10 | . . . 4 ⊢ (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹) | |
27 | 25, 26 | eqtri 2817 | . . 3 ⊢ ((𝐵 · 1) + 𝐷) = ((𝑇 · 1) + 𝐹) |
28 | 2, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27 | nummac 11981 | . 2 ⊢ ((𝑀 · 1) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
29 | 9, 28 | eqtr3i 2819 | 1 ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 (class class class)co 7007 1c1 10373 + caddc 10375 · cmul 10377 ℕ0cn0 11734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-ltxr 10515 df-sub 10708 df-nn 11476 df-n0 11735 |
This theorem is referenced by: decaddc 11991 |
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