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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 12127, 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 12087 | . 2 ⊢ ;(𝐴 + 1)0 = (((9 + 1) · (𝐴 + 1)) + 0) | |
2 | 9nn 11723 | . . . . . . . 8 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11637 | . . . . . . . 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 11637 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
7 | 5, 6 | nnmulcld 11678 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℕ) |
8 | 7 | nncnd 11641 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℂ) |
9 | 8 | addid1d 10829 | . . 3 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = ((9 + 1) · (𝐴 + 1))) |
10 | 4 | nncni 11635 | . . . . . 6 ⊢ (9 + 1) ∈ ℂ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℂ) |
12 | nncn 11633 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
13 | 1cnd 10625 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℂ) | |
14 | 11, 12, 13 | adddid 10654 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (((9 + 1) · 𝐴) + ((9 + 1) · 1))) |
15 | 11 | mulid1d 10647 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 1) = (9 + 1)) |
16 | 15 | oveq2d 7151 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (((9 + 1) · 𝐴) + (9 + 1))) |
17 | df-dec 12087 | . . . . . . 7 ⊢ ;𝐴9 = (((9 + 1) · 𝐴) + 9) | |
18 | 17 | oveq1i 7145 | . . . . . 6 ⊢ (;𝐴9 + 1) = ((((9 + 1) · 𝐴) + 9) + 1) |
19 | id 22 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ) | |
20 | 5, 19 | nnmulcld 11678 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℕ) |
21 | 20 | nncnd 11641 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℂ) |
22 | 2 | nncni 11635 | . . . . . . . 8 ⊢ 9 ∈ ℂ |
23 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → 9 ∈ ℂ) |
24 | 21, 23, 13 | addassd 10652 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((((9 + 1) · 𝐴) + 9) + 1) = (((9 + 1) · 𝐴) + (9 + 1))) |
25 | 18, 24 | syl5req 2846 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + (9 + 1)) = (;𝐴9 + 1)) |
26 | 16, 25 | eqtrd 2833 | . . . 4 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (;𝐴9 + 1)) |
27 | 14, 26 | eqtrd 2833 | . . 3 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (;𝐴9 + 1)) |
28 | 9, 27 | eqtrd 2833 | . 2 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = (;𝐴9 + 1)) |
29 | 1, 28 | syl5req 2846 | 1 ⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℕcn 11625 9c9 11687 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 |
This theorem is referenced by: (None) |
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