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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 12140, 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 12100 | . 2 ⊢ ;(𝐴 + 1)0 = (((9 + 1) · (𝐴 + 1)) + 0) | |
2 | 9nn 11736 | . . . . . . . 8 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11650 | . . . . . . . 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 11650 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
7 | 5, 6 | nnmulcld 11691 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℕ) |
8 | 7 | nncnd 11654 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℂ) |
9 | 8 | addid1d 10840 | . . 3 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = ((9 + 1) · (𝐴 + 1))) |
10 | 4 | nncni 11648 | . . . . . 6 ⊢ (9 + 1) ∈ ℂ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℂ) |
12 | nncn 11646 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
13 | 1cnd 10636 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℂ) | |
14 | 11, 12, 13 | adddid 10665 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (((9 + 1) · 𝐴) + ((9 + 1) · 1))) |
15 | 11 | mulid1d 10658 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 1) = (9 + 1)) |
16 | 15 | oveq2d 7172 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (((9 + 1) · 𝐴) + (9 + 1))) |
17 | df-dec 12100 | . . . . . . 7 ⊢ ;𝐴9 = (((9 + 1) · 𝐴) + 9) | |
18 | 17 | oveq1i 7166 | . . . . . 6 ⊢ (;𝐴9 + 1) = ((((9 + 1) · 𝐴) + 9) + 1) |
19 | id 22 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ) | |
20 | 5, 19 | nnmulcld 11691 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℕ) |
21 | 20 | nncnd 11654 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℂ) |
22 | 2 | nncni 11648 | . . . . . . . 8 ⊢ 9 ∈ ℂ |
23 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → 9 ∈ ℂ) |
24 | 21, 23, 13 | addassd 10663 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((((9 + 1) · 𝐴) + 9) + 1) = (((9 + 1) · 𝐴) + (9 + 1))) |
25 | 18, 24 | syl5req 2869 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + (9 + 1)) = (;𝐴9 + 1)) |
26 | 16, 25 | eqtrd 2856 | . . . 4 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (;𝐴9 + 1)) |
27 | 14, 26 | eqtrd 2856 | . . 3 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (;𝐴9 + 1)) |
28 | 9, 27 | eqtrd 2856 | . 2 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = (;𝐴9 + 1)) |
29 | 1, 28 | syl5req 2869 | 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 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℕcn 11638 9c9 11700 ;cdc 12099 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-dec 12100 |
This theorem is referenced by: (None) |
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