![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > numaddc | 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 9218 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2213 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
7 | 6 | nn0cni 9013 | . . . 4 ⊢ 𝑀 ∈ ℂ |
8 | 7 | mulid1i 7792 | . . 3 ⊢ (𝑀 · 1) = 𝑀 |
9 | 8 | oveq1i 5792 | . 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 9017 | . . 3 ⊢ 1 ∈ ℕ0 | |
14 | numaddc.8 | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
15 | 3 | nn0cni 9013 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
16 | 15 | mulid1i 7792 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
17 | 16 | oveq1i 5792 | . . . 4 ⊢ ((𝐴 · 1) + (𝐶 + 1)) = (𝐴 + (𝐶 + 1)) |
18 | 10 | nn0cni 9013 | . . . . 5 ⊢ 𝐶 ∈ ℂ |
19 | ax-1cn 7737 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | 15, 18, 19 | addassi 7798 | . . . 4 ⊢ ((𝐴 + 𝐶) + 1) = (𝐴 + (𝐶 + 1)) |
21 | numaddc.9 | . . . 4 ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 | |
22 | 17, 20, 21 | 3eqtr2i 2167 | . . 3 ⊢ ((𝐴 · 1) + (𝐶 + 1)) = 𝐸 |
23 | 4 | nn0cni 9013 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
24 | 23 | mulid1i 7792 | . . . . 5 ⊢ (𝐵 · 1) = 𝐵 |
25 | 24 | oveq1i 5792 | . . . 4 ⊢ ((𝐵 · 1) + 𝐷) = (𝐵 + 𝐷) |
26 | numaddc.10 | . . . 4 ⊢ (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹) | |
27 | 25, 26 | eqtri 2161 | . . 3 ⊢ ((𝐵 · 1) + 𝐷) = ((𝑇 · 1) + 𝐹) |
28 | 2, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27 | nummac 9250 | . 2 ⊢ ((𝑀 · 1) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
29 | 9, 28 | eqtr3i 2163 | 1 ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 (class class class)co 5782 1c1 7645 + caddc 7647 · cmul 7649 ℕ0cn0 9001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-inn 8745 df-n0 9002 |
This theorem is referenced by: decaddc 9260 |
Copyright terms: Public domain | W3C validator |