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Mirrors > Home > ILE Home > Th. List > numsucc | GIF version |
Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numsucc.1 | ⊢ 𝑌 ∈ ℕ0 |
numsucc.2 | ⊢ 𝑇 = (𝑌 + 1) |
numsucc.3 | ⊢ 𝐴 ∈ ℕ0 |
numsucc.4 | ⊢ (𝐴 + 1) = 𝐵 |
numsucc.5 | ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) |
Ref | Expression |
---|---|
numsucc | ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numsucc.2 | . . . . . . 7 ⊢ 𝑇 = (𝑌 + 1) | |
2 | numsucc.1 | . . . . . . . 8 ⊢ 𝑌 ∈ ℕ0 | |
3 | 1nn0 8993 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
4 | 2, 3 | nn0addcli 9014 | . . . . . . 7 ⊢ (𝑌 + 1) ∈ ℕ0 |
5 | 1, 4 | eqeltri 2212 | . . . . . 6 ⊢ 𝑇 ∈ ℕ0 |
6 | 5 | nn0cni 8989 | . . . . 5 ⊢ 𝑇 ∈ ℂ |
7 | 6 | mulid1i 7768 | . . . 4 ⊢ (𝑇 · 1) = 𝑇 |
8 | 7 | oveq2i 5785 | . . 3 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
9 | numsucc.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
10 | 9 | nn0cni 8989 | . . . 4 ⊢ 𝐴 ∈ ℂ |
11 | ax-1cn 7713 | . . . 4 ⊢ 1 ∈ ℂ | |
12 | 6, 10, 11 | adddii 7776 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
13 | 1 | eqcomi 2143 | . . . 4 ⊢ (𝑌 + 1) = 𝑇 |
14 | numsucc.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) | |
15 | 5, 9, 2, 13, 14 | numsuc 9195 | . . 3 ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝑇) |
16 | 8, 12, 15 | 3eqtr4ri 2171 | . 2 ⊢ (𝑁 + 1) = (𝑇 · (𝐴 + 1)) |
17 | numsucc.4 | . . 3 ⊢ (𝐴 + 1) = 𝐵 | |
18 | 17 | oveq2i 5785 | . 2 ⊢ (𝑇 · (𝐴 + 1)) = (𝑇 · 𝐵) |
19 | 9, 3 | nn0addcli 9014 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
20 | 17, 19 | eqeltrri 2213 | . . 3 ⊢ 𝐵 ∈ ℕ0 |
21 | 5, 20 | num0u 9192 | . 2 ⊢ (𝑇 · 𝐵) = ((𝑇 · 𝐵) + 0) |
22 | 16, 18, 21 | 3eqtri 2164 | 1 ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 ℕ0cn0 8977 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-n0 8978 |
This theorem is referenced by: decsucc 9222 |
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