Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12311 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
2 | decsplit0.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
3 | decsplit.4 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 1, 3 | nn0expcli 13661 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℕ0 |
5 | 2, 4 | nn0mulcli 12128 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℕ0 |
6 | 1, 5 | nn0mulcli 12128 | . . . . 5 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℕ0 |
7 | 6 | nn0cni 12102 | . . . 4 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℂ |
8 | decsplit.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
9 | 1, 8 | nn0mulcli 12128 | . . . . 5 ⊢ (;10 · 𝐵) ∈ ℕ0 |
10 | 9 | nn0cni 12102 | . . . 4 ⊢ (;10 · 𝐵) ∈ ℂ |
11 | decsplit.3 | . . . . 5 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 12102 | . . . 4 ⊢ 𝐷 ∈ ℂ |
13 | 7, 10, 12 | addassi 10843 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
14 | 1 | nn0cni 12102 | . . . . . 6 ⊢ ;10 ∈ ℂ |
15 | 5 | nn0cni 12102 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℂ |
16 | 8 | nn0cni 12102 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
17 | 14, 15, 16 | adddii 10845 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) |
18 | decsplit.6 | . . . . . 6 ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 | |
19 | 18 | oveq2i 7224 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = (;10 · 𝐶) |
20 | 17, 19 | eqtr3i 2767 | . . . 4 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) = (;10 · 𝐶) |
21 | 20 | oveq1i 7223 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · 𝐶) + 𝐷) |
22 | 13, 21 | eqtr3i 2767 | . 2 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) = ((;10 · 𝐶) + 𝐷) |
23 | decsplit.5 | . . . . . 6 ⊢ (𝑀 + 1) = 𝑁 | |
24 | 4 | nn0cni 12102 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℂ |
25 | 24, 14 | mulcomi 10841 | . . . . . 6 ⊢ ((;10↑𝑀) · ;10) = (;10 · (;10↑𝑀)) |
26 | 1, 3, 23, 25 | numexpp1 16631 | . . . . 5 ⊢ (;10↑𝑁) = (;10 · (;10↑𝑀)) |
27 | 26 | oveq2i 7224 | . . . 4 ⊢ (𝐴 · (;10↑𝑁)) = (𝐴 · (;10 · (;10↑𝑀))) |
28 | 2 | nn0cni 12102 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
29 | 28, 14, 24 | mul12i 11027 | . . . 4 ⊢ (𝐴 · (;10 · (;10↑𝑀))) = (;10 · (𝐴 · (;10↑𝑀))) |
30 | 27, 29 | eqtri 2765 | . . 3 ⊢ (𝐴 · (;10↑𝑁)) = (;10 · (𝐴 · (;10↑𝑀))) |
31 | dfdec10 12296 | . . 3 ⊢ ;𝐵𝐷 = ((;10 · 𝐵) + 𝐷) | |
32 | 30, 31 | oveq12i 7225 | . 2 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
33 | dfdec10 12296 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
34 | 22, 32, 33 | 3eqtr4i 2775 | 1 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7213 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 ℕ0cn0 12090 ;cdc 12293 ↑cexp 13635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-seq 13575 df-exp 13636 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |