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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 12142, 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 12102 | . 2 ⊢ ;(𝐴 + 1)0 = (((9 + 1) · (𝐴 + 1)) + 0) | |
2 | 9nn 11738 | . . . . . . . 8 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11652 | . . . . . . . 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 11652 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
7 | 5, 6 | nnmulcld 11693 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℕ) |
8 | 7 | nncnd 11656 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℂ) |
9 | 8 | addid1d 10842 | . . 3 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = ((9 + 1) · (𝐴 + 1))) |
10 | 4 | nncni 11650 | . . . . . 6 ⊢ (9 + 1) ∈ ℂ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℂ) |
12 | nncn 11648 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
13 | 1cnd 10638 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℂ) | |
14 | 11, 12, 13 | adddid 10667 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (((9 + 1) · 𝐴) + ((9 + 1) · 1))) |
15 | 11 | mulid1d 10660 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 1) = (9 + 1)) |
16 | 15 | oveq2d 7174 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (((9 + 1) · 𝐴) + (9 + 1))) |
17 | df-dec 12102 | . . . . . . 7 ⊢ ;𝐴9 = (((9 + 1) · 𝐴) + 9) | |
18 | 17 | oveq1i 7168 | . . . . . 6 ⊢ (;𝐴9 + 1) = ((((9 + 1) · 𝐴) + 9) + 1) |
19 | id 22 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ) | |
20 | 5, 19 | nnmulcld 11693 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℕ) |
21 | 20 | nncnd 11656 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℂ) |
22 | 2 | nncni 11650 | . . . . . . . 8 ⊢ 9 ∈ ℂ |
23 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → 9 ∈ ℂ) |
24 | 21, 23, 13 | addassd 10665 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((((9 + 1) · 𝐴) + 9) + 1) = (((9 + 1) · 𝐴) + (9 + 1))) |
25 | 18, 24 | syl5req 2871 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + (9 + 1)) = (;𝐴9 + 1)) |
26 | 16, 25 | eqtrd 2858 | . . . 4 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (;𝐴9 + 1)) |
27 | 14, 26 | eqtrd 2858 | . . 3 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (;𝐴9 + 1)) |
28 | 9, 27 | eqtrd 2858 | . 2 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = (;𝐴9 + 1)) |
29 | 1, 28 | syl5req 2871 | 1 ⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕcn 11640 9c9 11702 ;cdc 12101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-dec 12102 |
This theorem is referenced by: (None) |
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