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Mirrors > Home > MPE Home > Th. List > decsplit | Structured version Visualization version GIF version |
Description: Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
decsplit0.1 | ⊢ 𝐴 ∈ ℕ0 |
decsplit.2 | ⊢ 𝐵 ∈ ℕ0 |
decsplit.3 | ⊢ 𝐷 ∈ ℕ0 |
decsplit.4 | ⊢ 𝑀 ∈ ℕ0 |
decsplit.5 | ⊢ (𝑀 + 1) = 𝑁 |
decsplit.6 | ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 |
Ref | Expression |
---|---|
decsplit | ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 11516 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
2 | decsplit0.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
3 | decsplit.4 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 1, 3 | nn0expcli 12886 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℕ0 |
5 | 2, 4 | nn0mulcli 11331 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℕ0 |
6 | 1, 5 | nn0mulcli 11331 | . . . . 5 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℕ0 |
7 | 6 | nn0cni 11304 | . . . 4 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℂ |
8 | decsplit.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
9 | 1, 8 | nn0mulcli 11331 | . . . . 5 ⊢ (;10 · 𝐵) ∈ ℕ0 |
10 | 9 | nn0cni 11304 | . . . 4 ⊢ (;10 · 𝐵) ∈ ℂ |
11 | decsplit.3 | . . . . 5 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 11304 | . . . 4 ⊢ 𝐷 ∈ ℂ |
13 | 7, 10, 12 | addassi 10048 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
14 | 1 | nn0cni 11304 | . . . . . 6 ⊢ ;10 ∈ ℂ |
15 | 5 | nn0cni 11304 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℂ |
16 | 8 | nn0cni 11304 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
17 | 14, 15, 16 | adddii 10050 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) |
18 | decsplit.6 | . . . . . 6 ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 | |
19 | 18 | oveq2i 6661 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = (;10 · 𝐶) |
20 | 17, 19 | eqtr3i 2646 | . . . 4 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) = (;10 · 𝐶) |
21 | 20 | oveq1i 6660 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · 𝐶) + 𝐷) |
22 | 13, 21 | eqtr3i 2646 | . 2 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) = ((;10 · 𝐶) + 𝐷) |
23 | decsplit.5 | . . . . . 6 ⊢ (𝑀 + 1) = 𝑁 | |
24 | 4 | nn0cni 11304 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℂ |
25 | 24, 14 | mulcomi 10046 | . . . . . 6 ⊢ ((;10↑𝑀) · ;10) = (;10 · (;10↑𝑀)) |
26 | 1, 3, 23, 25 | numexpp1 15782 | . . . . 5 ⊢ (;10↑𝑁) = (;10 · (;10↑𝑀)) |
27 | 26 | oveq2i 6661 | . . . 4 ⊢ (𝐴 · (;10↑𝑁)) = (𝐴 · (;10 · (;10↑𝑀))) |
28 | 2 | nn0cni 11304 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
29 | 28, 14, 24 | mul12i 10231 | . . . 4 ⊢ (𝐴 · (;10 · (;10↑𝑀))) = (;10 · (𝐴 · (;10↑𝑀))) |
30 | 27, 29 | eqtri 2644 | . . 3 ⊢ (𝐴 · (;10↑𝑁)) = (;10 · (𝐴 · (;10↑𝑀))) |
31 | dfdec10 11497 | . . 3 ⊢ ;𝐵𝐷 = ((;10 · 𝐵) + 𝐷) | |
32 | 30, 31 | oveq12i 6662 | . 2 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
33 | dfdec10 11497 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
34 | 22, 32, 33 | 3eqtr4i 2654 | 1 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕ0cn0 11292 ;cdc 11493 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: (None) |
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