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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 12772 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
2 | decsplit0.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
3 | decsplit.4 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 1, 3 | nn0expcli 14135 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℕ0 |
5 | 2, 4 | nn0mulcli 12587 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℕ0 |
6 | 1, 5 | nn0mulcli 12587 | . . . . 5 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℕ0 |
7 | 6 | nn0cni 12561 | . . . 4 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℂ |
8 | decsplit.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
9 | 1, 8 | nn0mulcli 12587 | . . . . 5 ⊢ (;10 · 𝐵) ∈ ℕ0 |
10 | 9 | nn0cni 12561 | . . . 4 ⊢ (;10 · 𝐵) ∈ ℂ |
11 | decsplit.3 | . . . . 5 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 12561 | . . . 4 ⊢ 𝐷 ∈ ℂ |
13 | 7, 10, 12 | addassi 11296 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
14 | 1 | nn0cni 12561 | . . . . . 6 ⊢ ;10 ∈ ℂ |
15 | 5 | nn0cni 12561 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℂ |
16 | 8 | nn0cni 12561 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
17 | 14, 15, 16 | adddii 11298 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) |
18 | decsplit.6 | . . . . . 6 ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 | |
19 | 18 | oveq2i 7456 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = (;10 · 𝐶) |
20 | 17, 19 | eqtr3i 2764 | . . . 4 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) = (;10 · 𝐶) |
21 | 20 | oveq1i 7455 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · 𝐶) + 𝐷) |
22 | 13, 21 | eqtr3i 2764 | . 2 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) = ((;10 · 𝐶) + 𝐷) |
23 | decsplit.5 | . . . . . 6 ⊢ (𝑀 + 1) = 𝑁 | |
24 | 4 | nn0cni 12561 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℂ |
25 | 24, 14 | mulcomi 11294 | . . . . . 6 ⊢ ((;10↑𝑀) · ;10) = (;10 · (;10↑𝑀)) |
26 | 1, 3, 23, 25 | numexpp1 17120 | . . . . 5 ⊢ (;10↑𝑁) = (;10 · (;10↑𝑀)) |
27 | 26 | oveq2i 7456 | . . . 4 ⊢ (𝐴 · (;10↑𝑁)) = (𝐴 · (;10 · (;10↑𝑀))) |
28 | 2 | nn0cni 12561 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
29 | 28, 14, 24 | mul12i 11481 | . . . 4 ⊢ (𝐴 · (;10 · (;10↑𝑀))) = (;10 · (𝐴 · (;10↑𝑀))) |
30 | 27, 29 | eqtri 2762 | . . 3 ⊢ (𝐴 · (;10↑𝑁)) = (;10 · (𝐴 · (;10↑𝑀))) |
31 | dfdec10 12757 | . . 3 ⊢ ;𝐵𝐷 = ((;10 · 𝐵) + 𝐷) | |
32 | 30, 31 | oveq12i 7457 | . 2 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
33 | dfdec10 12757 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
34 | 22, 32, 33 | 3eqtr4i 2772 | 1 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 (class class class)co 7445 0cc0 11180 1c1 11181 + caddc 11183 · cmul 11185 ℕ0cn0 12549 ;cdc 12754 ↑cexp 14108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-seq 14049 df-exp 14109 |
This theorem is referenced by: (None) |
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