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Mirrors > Home > ILE Home > Th. List > numadd | GIF version |
Description: Add two decimal integers 𝑀 and 𝑁 (no 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) |
numadd.8 | ⊢ (𝐴 + 𝐶) = 𝐸 |
numadd.9 | ⊢ (𝐵 + 𝐷) = 𝐹 |
Ref | Expression |
---|---|
numadd | ⊢ (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 9187 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2210 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
7 | 6 | nn0cni 8982 | . . . 4 ⊢ 𝑀 ∈ ℂ |
8 | 7 | mulid1i 7761 | . . 3 ⊢ (𝑀 · 1) = 𝑀 |
9 | 8 | oveq1i 5777 | . 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 8986 | . . 3 ⊢ 1 ∈ ℕ0 | |
14 | 3 | nn0cni 8982 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
15 | 14 | mulid1i 7761 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
16 | 15 | oveq1i 5777 | . . . 4 ⊢ ((𝐴 · 1) + 𝐶) = (𝐴 + 𝐶) |
17 | numadd.8 | . . . 4 ⊢ (𝐴 + 𝐶) = 𝐸 | |
18 | 16, 17 | eqtri 2158 | . . 3 ⊢ ((𝐴 · 1) + 𝐶) = 𝐸 |
19 | 4 | nn0cni 8982 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
20 | 19 | mulid1i 7761 | . . . . 5 ⊢ (𝐵 · 1) = 𝐵 |
21 | 20 | oveq1i 5777 | . . . 4 ⊢ ((𝐵 · 1) + 𝐷) = (𝐵 + 𝐷) |
22 | numadd.9 | . . . 4 ⊢ (𝐵 + 𝐷) = 𝐹 | |
23 | 21, 22 | eqtri 2158 | . . 3 ⊢ ((𝐵 · 1) + 𝐷) = 𝐹 |
24 | 2, 3, 4, 10, 11, 1, 12, 13, 18, 23 | numma 9218 | . 2 ⊢ ((𝑀 · 1) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
25 | 9, 24 | eqtr3i 2160 | 1 ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5767 1c1 7614 + caddc 7616 · cmul 7618 ℕ0cn0 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-inn 8714 df-n0 8971 |
This theorem is referenced by: decadd 9228 |
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