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Mirrors > Home > MPE Home > Th. List > Mathboxes > deccarry | Structured version Visualization version GIF version |
Description: Add 1 to a 2 digit number with carry. This is a special case of decsucc 12714, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g., by applying this theorem three times we get (;;999 + 1) = ;;;1000. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.) (Proof shortened by AV, 10-Sep-2021.) |
Ref | Expression |
---|---|
deccarry | ⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 12674 | . 2 ⊢ ;(𝐴 + 1)0 = (((9 + 1) · (𝐴 + 1)) + 0) | |
2 | 9nn 12306 | . . . . . . . 8 ⊢ 9 ∈ ℕ | |
3 | peano2nn 12220 | . . . . . . . 8 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ (9 + 1) ∈ ℕ |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℕ) |
6 | peano2nn 12220 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
7 | 5, 6 | nnmulcld 12261 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℕ) |
8 | 7 | nncnd 12224 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℂ) |
9 | 8 | addridd 11410 | . . 3 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = ((9 + 1) · (𝐴 + 1))) |
10 | 4 | nncni 12218 | . . . . . 6 ⊢ (9 + 1) ∈ ℂ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℂ) |
12 | nncn 12216 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
13 | 1cnd 11205 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℂ) | |
14 | 11, 12, 13 | adddid 11234 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (((9 + 1) · 𝐴) + ((9 + 1) · 1))) |
15 | 11 | mulridd 11227 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 1) = (9 + 1)) |
16 | 15 | oveq2d 7420 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (((9 + 1) · 𝐴) + (9 + 1))) |
17 | df-dec 12674 | . . . . . . 7 ⊢ ;𝐴9 = (((9 + 1) · 𝐴) + 9) | |
18 | 17 | oveq1i 7414 | . . . . . 6 ⊢ (;𝐴9 + 1) = ((((9 + 1) · 𝐴) + 9) + 1) |
19 | id 22 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ) | |
20 | 5, 19 | nnmulcld 12261 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℕ) |
21 | 20 | nncnd 12224 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℂ) |
22 | 2 | nncni 12218 | . . . . . . . 8 ⊢ 9 ∈ ℂ |
23 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → 9 ∈ ℂ) |
24 | 21, 23, 13 | addassd 11232 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((((9 + 1) · 𝐴) + 9) + 1) = (((9 + 1) · 𝐴) + (9 + 1))) |
25 | 18, 24 | eqtr2id 2786 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + (9 + 1)) = (;𝐴9 + 1)) |
26 | 16, 25 | eqtrd 2773 | . . . 4 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (;𝐴9 + 1)) |
27 | 14, 26 | eqtrd 2773 | . . 3 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (;𝐴9 + 1)) |
28 | 9, 27 | eqtrd 2773 | . 2 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = (;𝐴9 + 1)) |
29 | 1, 28 | eqtr2id 2786 | 1 ⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 (class class class)co 7404 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ℕcn 12208 9c9 12270 ;cdc 12673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-dec 12674 |
This theorem is referenced by: (None) |
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